Acoustic correlates of perceived prominence in German and Catalan: Bayesian Modeling
Aleksandra Ćwiek, Alina Gregori, Paula G. Sánchez-Ramón, Frank Kügler, Pilar Prieto
27 November, 2024
1 Introduction
This is the analysis using Bayesian Modeling.
2 Data preparation
2.1 Source setup
########## folders ##########
# current folder (first go to session -> set working directory -> to source file location)
parentfolder <- dirname(getwd())
data <- paste0(parentfolder, '/MultIS_data/')
audiodata <- paste0(parentfolder, '/audio_processed/')
syllables <- paste0(audiodata, 'syllables/')
dataworkspace <- paste0(parentfolder, '/data_processed/')
datamerged <- paste0(parentfolder, '/data_merged/')
datasets <- paste0(parentfolder, '/datasets/')
models <- paste0(parentfolder, '/models/')
plots <- paste0(parentfolder, '/plots/')
scripts <- paste0(parentfolder, '/scripts/')
########## source file ##########
#source(paste0(scripts, "adjectives-preparation.R"))
#################### packages ####################
# Data Manipulation
library(tibble)
library(stringr)
library(tidyverse) # includes readr, tidyr, dplyr, ggplot2
packageVersion("tidyverse")## [1] '2.0.0'library(data.table)
# Plotting
library(ggforce)
library(ggpubr)
library(gridExtra)
library(corrplot)
# Bayesian
library(brms); packageVersion("brms")## [1] '2.22.0'library(cmdstanr); packageVersion("cmdstanr")## [1] '0.8.1'library(emmeans); packageVersion("emmeans")## [1] '1.10.5'library(posterior); packageVersion("posterior")## [1] '1.6.0'# tidybayes https://mjskay.github.io/tidybayes/articles/tidy-brms.html
library(magrittr)
library(dplyr)
library(purrr)
library(forcats)
library(tidyr)
library(modelr)
library(ggdist)
library(tidybayes)
library(ggplot2)
library(cowplot)
library(rstan)
library(brms)
library(ggrepel)
library(RColorBrewer)
library(gganimate)
library(posterior)
library(distributional)
theme_set(theme_tidybayes() + panel_border())
# use all available cores for parallel computing
options(mc.cores = parallel::detectCores())
colorBlindBlack8 <- c("#000000", "#E69F00", "#56B4E9", "#009E73",
"#F0E442", "#0072B2", "#D55E00", "#CC79A7")2.2 Functions
# Function to back-transform fixed effects from a model
backtransform_fixef <- function(model) {
# Extract fixed effects (population-level)
fixef_output <- fixef(model)
# Extract the estimates for the intercept and percProm levels
intercept <- fixef_output["Intercept", "Estimate"]
percProm1 <- fixef_output["percProm1", "Estimate"]
percProm2 <- fixef_output["percProm2", "Estimate"]
percProm3 <- fixef_output["percProm3", "Estimate"]
# Combine the effects for each percProm level before applying exp()
combined_effects <- data.frame(
percProm_level = c("percProm0", "percProm1", "percProm2", "percProm3"),
Estimate = exp(c(intercept, # percProm0 is just the intercept
intercept + percProm1, # Add percProm1 effect to intercept
intercept + percProm2, # Add percProm2 effect to intercept
intercept + percProm3)), # Add percProm3 effect to intercept
# You can calculate confidence intervals by combining Q2.5 and Q97.5 as well
Q2.5 = exp(c(fixef_output["Intercept", "Q2.5"], # percProm0 interval
fixef_output["Intercept", "Q2.5"] + fixef_output["percProm1", "Q2.5"], # percProm1 interval
fixef_output["Intercept", "Q2.5"] + fixef_output["percProm2", "Q2.5"], # percProm2 interval
fixef_output["Intercept", "Q2.5"] + fixef_output["percProm3", "Q2.5"])), # percProm3 interval
Q97.5 = exp(c(fixef_output["Intercept", "Q97.5"],
fixef_output["Intercept", "Q97.5"] + fixef_output["percProm1", "Q97.5"],
fixef_output["Intercept", "Q97.5"] + fixef_output["percProm2", "Q97.5"],
fixef_output["Intercept", "Q97.5"] + fixef_output["percProm3", "Q97.5"]))
)
return(combined_effects)
}
combine_fixed_random_backtransforming <- function(participant_id) {
# Get the random effects for the given participant
rand_intercept <- ranef_output$participant[participant_id, , "Intercept"]
rand_percProm1 <- ranef_output$participant[participant_id, , "percProm1"]
rand_percProm2 <- ranef_output$participant[participant_id, , "percProm2"]
rand_percProm3 <- ranef_output$participant[participant_id, , "percProm3"]
# Combine fixed and random effects
participant_effects <- data.frame(
percProm_level = c("percProm0", "percProm1", "percProm2", "percProm3"),
Estimate = exp(c(
intercept + rand_intercept["Estimate"],
intercept + percProm0 + rand_intercept["Estimate"] + rand_percProm1["Estimate"],
intercept + percProm2 + rand_intercept["Estimate"] + rand_percProm2["Estimate"],
intercept + percProm3 + rand_intercept["Estimate"] + rand_percProm3["Estimate"]
)),
Est.Error = c(
sqrt(fixef_output["Intercept", "Est.Error"]^2 + rand_intercept["Est.Error"]^2),
sqrt(
fixef_output["Intercept", "Est.Error"]^2 + fixef_output["percProm1", "Est.Error"]^2 +
rand_intercept["Est.Error"]^2 + rand_percProm1["Est.Error"]^2
),
sqrt(
fixef_output["Intercept", "Est.Error"]^2 + fixef_output["percProm2", "Est.Error"]^2 +
rand_intercept["Est.Error"]^2 + rand_percProm2["Est.Error"]^2
),
sqrt(
fixef_output["Intercept", "Est.Error"]^2 + fixef_output["percProm3", "Est.Error"]^2 +
rand_intercept["Est.Error"]^2 + rand_percProm3["Est.Error"]^2
)
),
Q2.5 = exp(c(
intercept + rand_intercept["Q2.5"],
intercept + percProm0 + rand_intercept["Q2.5"] + rand_percProm1["Q2.5"],
intercept + percProm2 + rand_intercept["Q2.5"] + rand_percProm2["Q2.5"],
intercept + percProm3 + rand_intercept["Q2.5"] + rand_percProm3["Q2.5"]
)),
Q97.5 = exp(c(
intercept + rand_intercept["Q97.5"],
intercept + percProm0 + rand_intercept["Q97.5"] + rand_percProm1["Q97.5"],
intercept + percProm2 + rand_intercept["Q97.5"] + rand_percProm2["Q97.5"],
intercept + percProm3 + rand_intercept["Q97.5"] + rand_percProm3["Q97.5"]
))
)
participant_effects$participant <- participant_id
return(participant_effects)
}
combine_fixed_random_student <- function(participant_id) {
# Get the random effects for the given participant
rand_intercept <- ranef_output$participant[participant_id, , "Intercept"]
rand_percProm1 <- ranef_output$participant[participant_id, , "percProm1"]
rand_percProm2 <- ranef_output$participant[participant_id, , "percProm2"]
rand_percProm3 <- ranef_output$participant[participant_id, , "percProm3"]
# Combine fixed and random effects for each percProm level
participant_effects <- data.frame(
percProm_level = c("percProm0", "percProm1", "percProm2", "percProm3"),
# Estimate: sum of fixed and random effects
Estimate = c(
rand_intercept["Estimate"],
rand_intercept["Estimate"] + rand_percProm1["Estimate"],
rand_intercept["Estimate"] + rand_percProm2["Estimate"],
rand_intercept["Estimate"] + rand_percProm3["Estimate"]
),
# Standard Error: combining errors using sqrt of sum of squares
Est.Error = c(
sqrt(rand_intercept["Est.Error"]^2),
sqrt(rand_intercept["Est.Error"]^2 + rand_percProm1["Est.Error"]^2),
sqrt(rand_intercept["Est.Error"]^2 + rand_percProm2["Est.Error"]^2),
sqrt(rand_intercept["Est.Error"]^2 + rand_percProm3["Est.Error"]^2)
),
# Q2.5 and Q97.5: these need to be simulated to reflect the full posterior distribution
Q2.5 = c(
rand_intercept["Q2.5"],
rand_intercept["Q2.5"] + rand_percProm1["Q2.5"],
rand_intercept["Q2.5"] + rand_percProm2["Q2.5"],
rand_intercept["Q2.5"] + rand_percProm3["Q2.5"]
),
Q97.5 = c(
rand_intercept["Q97.5"],
rand_intercept["Q97.5"] + rand_percProm1["Q97.5"],
rand_intercept["Q97.5"] + rand_percProm2["Q97.5"],
rand_intercept["Q97.5"] + rand_percProm3["Q97.5"]
)
)
participant_effects$participant <- participant_id
return(participant_effects)
}
# Function to compute summary statistics for each difference
compute_summary <- function(diff) {
estimate <- mean(diff)
se <- sd(diff)
ci_lower <- quantile(diff, 0.025)
ci_upper <- quantile(diff, 0.975)
prob <- mean(diff > 0) # Posterior probability that difference > 0
return(data.frame(Estimate = estimate, SE = se, CI.Lower = ci_lower, CI.Upper = ci_upper, Prob = prob))
}2.3 Load in data frames
participant_info <- read_delim(paste0(data,"ParticipantInfo_GERCAT.csv"), delim = ";")
# Load the information about duration of each segment (if needed)
data_df <- read.table(paste0(syllables, "fileDurationsDF.csv"), header = TRUE, sep = ',')
# Load cleaned syllable data
data <- read_csv(paste0(datasets, "data_cleaned.csv"))
# Load cleaned targets data
targets <- read_csv(paste0(datasets, "targets.csv"))
# Load cleaned targets with pre-post data
data_prepost <- read_csv(paste0(datasets, "data_prepost.csv"))2.4 You can add participant info
# Process participant_info so that participant number column is only number
participant_info$Participant <- parse_number(participant_info$Participant)
# Convert the column names of participant_info to lowercase
colnames(participant_info) <- tolower(colnames(participant_info))
# Merge the dataframes by "Participant" and "Language"
data_prepost <- merge(data_prepost, participant_info, by = c("participant", "language"), all.x = TRUE)3 Data preparation
We split the table in two languages. Then, we only keep the relevant columns (top 10 features).
# Turn variables to factors
data_prepost$percProm <- as.factor(data_prepost$percProm)
data_prepost$itemNum <- as.factor(data_prepost$itemNum)
data_prepost$focus <- as.factor(data_prepost$focus)
data_prepost$participant <- as.factor(data_prepost$participant)
# Add contrast-coded gender info (in case we want to use if)
data_prepost <- data_prepost %>%
mutate(gender_s = case_when(gender == "female" ~ 0.5,
gender == "male" ~ -0.5))
# First, remove some columns for both languages
data_prepost <- data_prepost %>%
select(-f1_freq_median, -f1_freq_median_norm, -f2_freq_median, -f2_freq_median_norm,
-f1_freq_medianPre, -f1_freq_median_normPre, -f2_freq_medianPre, -f2_freq_median_normPre,
-f1_freq_medianPost, -f1_freq_median_normPost, -f2_freq_medianPost, -f2_freq_median_normPost
)
# Adapt duration scale (because we set the priors for ms, not s)
data_prepost$duration <- data_prepost$duration * 1000
data_prepost$durationPre <- data_prepost$durationPre * 1000
data_prepost$durationPost <- data_prepost$durationPost * 1000
data_prepost$duration_noSilence <- data_prepost$duration_noSilence * 1000
data_prepost$duration_noSilencePre <- data_prepost$duration_noSilencePre * 1000
data_prepost$duration_noSilencePost <- data_prepost$duration_noSilencePost * 1000
# Create data_prepost_german for rows where language is German
data_prepost_ger <- data_prepost %>%
filter(language == "German")
# Create data_prepost_catalan for rows where language is Catalan
data_prepost_cat <- data_prepost %>%
filter(language == "Catalan")
# Select columns for German
data_prepost_ger <- data_prepost_ger %>%
select(fileName, language, participant, gender, gender_s, age,
itemNum, focus, annotationNum, annotationNumTarget,
word, syllText, syllTextPre, syllTextPost, percProm,
f0_slope, f0_slope_norm,
pitch_median, pitch_median_norm,
ampl_sd,
ampl_noSilence_medianPost,
ampl_noSilence_median,
duration,
flux_sd,
flux_median,
ampl_noSilence_sd,
pitch_medianPost, pitch_median_normPost)
# Select columns for Catalan
data_prepost_cat <- data_prepost_cat %>%
select(fileName, language, participant, gender, gender_s, age,
itemNum, focus, annotationNum, annotationNumTarget,
word, syllText, syllTextPre, syllTextPost, percProm,
duration,
f0_slopePost, f0_slope_normPost,
fmDep_medianPost,
entropySh_sd,
flux_medianPost,
ampl_medianPost,
specCentroid_median,
duration_noSilence,
durationPre,
ampl_sd)Add information on pauses to test separately. First in German.
# First, let's create a copy of data_prepost_ger to work on
data_prepost_ger_updated <- data_prepost_ger
# Add a new column durationPause and pausePre initialized to NA
data_prepost_ger_updated$durationPause <- NA
data_prepost_ger_updated$pausePre <- NA
# Loop through each row in data_prepost_ger where syllTextPre is NA
for (i in 1:nrow(data_prepost_ger_updated)) {
# Check if syllTextPre is NA
if (is.na(data_prepost_ger_updated$syllTextPre[i])) {
# Get the corresponding fileName in this row
fileName_i <- data_prepost_ger_updated$fileName[i]
# Find the index of this fileName in the data dataset
match_idx <- which(data$fileName == fileName_i)
# If a match is found, and the previous row exists (to avoid out of bounds)
if (length(match_idx) > 0 && match_idx > 1) {
# Look at the previous row's syllText to see if it's "-p-"
if (data$syllText[match_idx - 1] == "-p-") {
# Extract the duration value from the previous row
duration_value <- data$duration[match_idx - 1]
# Update syllTextPre with "-p-" in data_prepost_ger
data_prepost_ger_updated$syllTextPre[i] <- "-p-"
# Update the durationPause column with the value from data
data_prepost_ger_updated$durationPause[i] <- duration_value
}
}
}
}
rm(i, fileName_i, duration_value, match_idx)
# Create the pausePre column based on whether durationPause is NA or not
data_prepost_ger_updated$pausePre <- ifelse(is.na(data_prepost_ger_updated$durationPause), "no", "yes")
data_prepost_ger <- data_prepost_ger_updated
rm(data_prepost_ger_updated)Inspect.
table(data_prepost_ger$pausePre)##
## no yes
## 2341 3prop.table(table(data_prepost_ger$pausePre))##
## no yes
## 0.998720137 0.001279863It does not really make sense to investigate this in German.
Now in Catalan.
# First, let's create a copy of data_prepost_cat to work on
data_prepost_cat_updated <- data_prepost_cat
# Add a new column durationPause and pausePre initialized to NA
data_prepost_cat_updated$durationPause <- NA
data_prepost_cat_updated$pausePre <- NA
# Loop through each row in data_prepost_cat where syllTextPre is NA
for (i in 1:nrow(data_prepost_cat_updated)) {
# Check if syllTextPre is NA
if (is.na(data_prepost_cat_updated$syllTextPre[i])) {
# Get the corresponding fileName in this row
fileName_i <- data_prepost_cat_updated$fileName[i]
# Find the index of this fileName in the data dataset
match_idx <- which(data$fileName == fileName_i)
# If a match is found, and the previous row exists (to avoid out of bounds)
if (length(match_idx) > 0 && match_idx > 1) {
# Look at the previous row's syllText to see if it's "-p-"
if (data$syllText[match_idx - 1] == "-p-") {
# Extract the duration value from the previous row
duration_value <- data$duration[match_idx - 1]
# Update syllTextPre with "-p-" in data_prepost_cat
data_prepost_cat_updated$syllTextPre[i] <- "-p-"
# Update the durationPause column with the value from data
data_prepost_cat_updated$durationPause[i] <- duration_value
}
}
}
}
rm(i, fileName_i, duration_value, match_idx)
# Create the pausePre column based on whether durationPause is NA or not
data_prepost_cat_updated$pausePre <- ifelse(is.na(data_prepost_cat_updated$durationPause), "no", "yes")
data_prepost_cat <- data_prepost_cat_updated
rm(data_prepost_cat_updated)What are the values in Catalan?
table(data_prepost_cat$pausePre)##
## no yes
## 2148 98prop.table(table(data_prepost_cat$pausePre))##
## no yes
## 0.95636687 0.04363313It makes more sense.
4 Modeling
Given the goal of comparing the thresholds of changes in acoustic features across prominence levels, while accounting for speaker variability differences, a hierarchical (or mixed-effects) Bayesian model with acoustic features as outcome variables will be a suitable approach. This model allows you to account for the variability across participants while focusing on estimating the thresholds.
4.1 Catalan
Let’s check the correlations between acoustic features.
# Calculate the correlation matrix
corrCat <- cor(data_prepost_cat[, c("duration", "f0_slopePost", "fmDep_medianPost",
"entropySh_sd", "flux_medianPost",
"ampl_medianPost", "specCentroid_median", "duration_noSilence",
"durationPre", "ampl_sd")], use = "complete.obs")
corrplot(corrCat, method = "color", tl.cex = 0.7, number.cex = 0.7, addCoef.col = "black")
Let’s set contrasts for comparisons.
# Drop unused levels of percProm
data_prepost_cat$percProm <- droplevels(data_prepost_cat$percProm)
# Set sum contrasts for percProm to compare all levels
#contrasts(data_prepost_cat$percProm) <- contr.sum(length(unique(data_prepost_cat$percProm)))What is the mean for gender.
summary(data_prepost_cat$gender_s)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.5000 -0.5000 0.5000 0.1563 0.5000 0.5000# 0.1563We go one by one the ten features from top to bottom.
4.1.1 Cumulative
data_prepost_cat$percProm <- factor(data_prepost_cat$percProm,
levels = c(1, 2, 3),
ordered = TRUE)4.1.1.1 Formulas
durCum <- bf(
percProm ~
duration +
f0_slopePost +
fmDep_medianPost +
entropySh_sd +
flux_medianPost +
ampl_medianPost +
specCentroid_median +
durationPre +
ampl_sd +
(1 | participant)
)
durCum_ranSlopes <- bf(
percProm ~
duration +
f0_slopePost +
fmDep_medianPost +
entropySh_sd +
flux_medianPost +
ampl_medianPost +
specCentroid_median +
durationPre +
ampl_sd +
(1 + duration +
f0_slopePost +
fmDep_medianPost +
flux_medianPost +
ampl_medianPost +
specCentroid_median +
durationPre +
ampl_sd | participant)
)
durCum_allInt <- bf(
percProm ~
(duration +
f0_slopePost +
fmDep_medianPost +
entropySh_sd +
flux_medianPost +
ampl_medianPost +
specCentroid_median +
durationPre +
ampl_sd) *
(duration +
f0_slopePost +
fmDep_medianPost +
flux_medianPost +
ampl_medianPost +
specCentroid_median +
durationPre +
ampl_sd) +
(1 + duration +
f0_slopePost +
fmDep_medianPost +
flux_medianPost +
ampl_medianPost +
specCentroid_median +
durationPre +
ampl_sd | participant)
)4.1.1.2 Priors
4.1.1.3 Model
pp_check(mdl_durCumCat)
conditional_effects(mdl_durCumCat, re_formula = NA, categorical = TRUE)
hypothesis(mdl_durCumCat, c("duration = 0",
"f0_slopePost = 0",
"fmDep_medianPost = 0",
"entropySh_sd = 0",
"flux_medianPost = 0",
"ampl_medianPost = 0",
"specCentroid_median = 0",
"durationPre = 0",
"ampl_sd = 0"))## Hypothesis Tests for class b:
## Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio
## 1 (duration) = 0 0.01 0.00 0.01 0.01 NA
## 2 (f0_slopePost) = 0 -0.66 0.18 -1.02 -0.30 NA
## 3 (fmDep_medianPost) = 0 -0.01 0.11 -0.22 0.20 NA
## 4 (entropySh_sd) = 0 1.23 1.65 -2.01 4.50 NA
## 5 (flux_medianPost) = 0 3.35 1.69 0.01 6.66 NA
## 6 (ampl_medianPost) = 0 1.65 0.91 -0.14 3.43 NA
## 7 (specCentroid_med... = 0 0.00 0.00 -0.01 0.00 NA
## 8 (durationPre) = 0 0.00 0.00 0.00 0.01 NA
## 9 (ampl_sd) = 0 1.92 1.60 -1.21 5.10 NA
## Post.Prob Star
## 1 NA *
## 2 NA *
## 3 NA
## 4 NA
## 5 NA *
## 6 NA
## 7 NA *
## 8 NA *
## 9 NA
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.4.1.2 Duration
4.1.2.1 Priors
4.1.2.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1724661 0.01381479 0.1460578 0.1997325An R² of approximately 0.17 suggests a modest
fit. This value indicates that the model explains about
17.2% of the variance in the dependent variable
duration. In other words, a small portion of the
variability in duration is accounted for by the predictors
in the model. The 95% credible interval, ranging from
14.6% to 19.9%, shows some uncertainty around this
estimate but provides a stable indication of the model’s limited
explanatory power.
4.1.2.3 Fixed effects
Let’s check how it looks like.
# Summarize the model
summary(mdl_durationCat)## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: duration ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 2246)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 0.16 0.03 0.11 0.23 1.00 7165
## sd(percProm2) 0.14 0.02 0.11 0.18 1.00 6430
## sd(percProm3) 0.18 0.04 0.12 0.27 1.00 7623
## cor(percProm1,percProm2) 0.66 0.15 0.32 0.90 1.00 4310
## cor(percProm1,percProm3) 0.47 0.23 -0.02 0.85 1.00 4645
## cor(percProm2,percProm3) 0.68 0.15 0.32 0.92 1.00 6921
## Tail_ESS
## sd(percProm1) 9788
## sd(percProm2) 9675
## sd(percProm3) 9345
## cor(percProm1,percProm2) 6261
## cor(percProm1,percProm3) 7693
## cor(percProm2,percProm3) 9172
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 5.11 0.03 5.04 5.18 1.00 6387 8711
## percProm2 5.22 0.03 5.17 5.27 1.00 6428 9582
## percProm3 5.35 0.04 5.27 5.43 1.00 8611 11633
## gender_s 0.06 0.04 0.00 0.15 1.00 6584 8062
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.33 0.01 0.32 0.34 1.00 23416 11411
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).# Extract fixed effects
exp(fixef(mdl_durationCat))## Estimate Est.Error Q2.5 Q97.5
## percProm1 165.351091 1.034530 154.740478 176.985674
## percProm2 184.973169 1.026270 175.846273 194.738125
## percProm3 211.451562 1.041156 194.961916 228.371297
## gender_s 1.064828 1.041079 1.003856 1.166572It shows a stable increase from 1 to 3.
Diagnostic plots.
plot(mdl_durationCat)
Posterior predictive check.
pp_check(mdl_durationCat, ndraws = 100)
pp_check(mdl_durationCat, type = "error_scatter_avg", ndraws = 100)
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 56 144 180 201 238 7094.1.2.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 165 155 177
## 2 185 176 195
## 3 212 195 228
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 -19.6 -29.7 -9.47
## percProm1 - percProm3 -46.2 -62.9 -29.25
## percProm2 - percProm3 -26.6 -40.5 -12.20
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.2.3.2 Hypothesis testing
diff_1_2: Estimate = -19.6 [-29.6, -9.36], Posterior Probability = 99.9%; Strong evidence suggests that thedurationatpercProm2is higher than atpercProm1. The credible interval does not include zero, and the high posterior probability confirms this difference.diff_1_3: Estimate = -46.2 [-62.9, -29.3], Posterior Probability = 100%; Strong evidence indicates that thedurationatpercProm3is higher than atpercProm1. The negative estimate and credible interval entirely below zero show a reliable difference between these levels.diff_2_3: Estimate = -26.6 [-40.8, -12.5], Posterior Probability = 100%; Strong evidence thatdurationatpercProm3is higher than atpercProm2. The credible interval is entirely negative, supporting this increase indurationwith risingpercProm.
Overall Implications:
Trend: The results indicate a consistent increase in
durationacrosspercPromlevels, moving frompercProm1topercProm3.Interpretation: Each higher level of
percPromis associated with a reliably longerduration. The strong posterior probabilities and credible intervals suggest that as perceived prominence (percProm) increases, so doesduration, supporting a stable trend in the data.
4.1.2.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.2.4 Group-level effects
First, get a general overview.
percProm1: The estimated standard deviation is 0.160 (95% CrI: 0.108 to 0.225), indicating moderate variability in participants’ response to
percProm1.percProm2: The estimated standard deviation is 0.138 (95% CrI: 0.105 to 0.182), suggesting slightly less variability in participants’ response to
percProm2compared topercProm1.percProm3: The estimated standard deviation is 0.184 (95% CrI: 0.119 to 0.266), showing increased variability in response to
percProm3relative to the other prominence levels, indicating that participants exhibit a wider range of responses at this level.
Overall Interpretation: These results suggest that
as perceived prominence increases from percProm1 to
percProm3, the variability in participants’ responses tends
to fluctuate, with percProm3 showing the highest
variability. This pattern may imply that the highest level of perceived
prominence elicits a more diverse range of responses in terms of
durationCat.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.3 F0 slope (post-tonic)
4.1.3.1 Priors
4.1.3.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.06645816 0.007244046 0.05251865 0.08075153An R² of approximately 0.07 suggests a weak fit,
indicating that the model explains only about 6.6% of the
variance in the dependent variable
f0_slopePostCat. In other words, a small portion of the
variability in f0_slopePostCat is accounted for by the
predictors in the model. The 95% credible interval, ranging from
5.3% to 8.1%, reflects some uncertainty around this
estimate, but it generally confirms the model’s limited explanatory
power.
4.1.3.3 Fixed effects
Let’s check how it looks like.
# Summarize the model
summary(mdl_f0_slopePostCat)## Family: student
## Links: mu = identity; sigma = identity; nu = identity
## Formula: f0_slopePost ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 1780)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 0.03 0.02 0.00 0.06 1.00 3199
## sd(percProm2) 0.08 0.01 0.06 0.11 1.00 6465
## sd(percProm3) 0.09 0.02 0.05 0.14 1.00 6050
## cor(percProm1,percProm2) 0.15 0.41 -0.73 0.85 1.00 691
## cor(percProm1,percProm3) 0.16 0.43 -0.72 0.88 1.00 1503
## cor(percProm2,percProm3) 0.36 0.24 -0.18 0.76 1.00 10503
## Tail_ESS
## sd(percProm1) 6030
## sd(percProm2) 9786
## sd(percProm3) 6791
## cor(percProm1,percProm2) 1116
## cor(percProm1,percProm3) 2586
## cor(percProm2,percProm3) 11607
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 0.00 0.01 -0.02 0.03 1.00 7434 10709
## percProm2 0.06 0.02 0.03 0.09 1.00 7191 9945
## percProm3 -0.08 0.02 -0.12 -0.03 1.00 10715 11007
## gender_s -0.01 0.02 -0.05 0.03 1.00 9400 9275
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.13 0.01 0.12 0.14 1.00 12273 12072
## nu 2.00 0.14 1.75 2.29 1.00 14263 11993
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).# Extract fixed effects
fixef(mdl_f0_slopePostCat)## Estimate Est.Error Q2.5 Q97.5
## percProm1 0.004748638 0.01197570 -0.01867902 0.02921571
## percProm2 0.062793941 0.01593530 0.03176724 0.09456290
## percProm3 -0.078127142 0.02124579 -0.11879674 -0.03435458
## gender_s -0.010273064 0.01895841 -0.04863810 0.02640334It shows a stable increase from 1 to 2, and then a drop from 2 to 3.
Diagnostic plots.
plot(mdl_f0_slopePostCat)
Posterior predictive check.
pp_check(mdl_f0_slopePostCat, ndraws = 100)
pp_check(mdl_f0_slopePostCat, type = "error_scatter_avg", ndraws = 100)
Extract samples.
# Extract posterior samples of fixed effects
samples_mdl_f0_slopePostCat <- as_draws_array(mdl_f0_slopePostCat,
variable = "^b_",
regex = TRUE)
# Summarize posterior draws
summarize_draws(samples_mdl_f0_slopePostCat)Check conditional effect.
# Plot conditional effects (fixed effects only)
conditional_effects(mdl_f0_slopePostCat, re_formula = NA)
# Extract conditional effects data
conditional_effects(mdl_f0_slopePostCat, plot = FALSE, re_formula = NA)$percPromCompare conditional effects to raw values.
# Calculate raw means by percProm level
data_prepost_cat %>%
group_by(percProm) %>%
summarize(avg_f0_slopePost = mean(f0_slopePost, na.rm = TRUE))# Summary of f0_slopePost
summary(data_prepost_cat$f0_slopePost)## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## -4.0920 -0.0858 0.0052 0.0324 0.1774 4.2460 4664.1.3.3.1 Effect comparison
# Compute estimated marginal means on log-scale
em_mdl_f0_slopePostCat <- emmeans(mdl_f0_slopePostCat, ~ percProm)
print(em_mdl_f0_slopePostCat)## percProm emmean lower.HPD upper.HPD
## 1 0.00453 -0.0188 0.0291
## 2 0.06272 0.0315 0.0942
## 3 -0.07870 -0.1201 -0.0358
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95plot(em_mdl_f0_slopePostCat)
# Perform pairwise comparisons between levels of percProm
emPairs_mdl_f0_slopePostCat <- pairs(emmeans(mdl_f0_slopePostCat, ~ percProm))
print(emPairs_mdl_f0_slopePostCat)## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 -0.0578 -0.0949 -0.0209
## percProm1 - percProm3 0.0832 0.0383 0.1305
## percProm2 - percProm3 0.1412 0.0952 0.1861
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95plot(emPairs_mdl_f0_slopePostCat)
If the comparison includes 0, the contrast is not reliably different.
4.1.3.3.2 Hypothesis testing
# Extract posterior samples of the fixed effects coefficients
as_draws_df(mdl_f0_slopePostCat) %>%
# Select fixed effects for percProm levels
select(starts_with("b_percProm")) %>%
# Rename columns to match percProm levels
rename_with(~ str_replace(.x, "b_percProm", "percProm"), starts_with("b_percProm")) %>%
# Compute differences between percProm levels
mutate(
diff_1_2 = percProm1 - percProm2,
diff_1_3 = percProm1 - percProm3,
diff_2_3 = percProm2 - percProm3
) %>%
# Gather differences into long format
pivot_longer(
cols = starts_with("diff_"),
names_to = "Comparison",
values_to = "Difference"
) %>%
# Group by comparison to compute summary statistics
group_by(Comparison) %>%
summarise(
Estimate = mean(Difference),
Est.Error = sd(Difference),
CI.Lower = quantile(Difference, 0.025),
CI.Upper = quantile(Difference, 0.975),
Post.Prob = if_else(Estimate > 0,
mean(Difference > 0) * 100,
mean(Difference < 0) * 100)
) %>%
ungroup() %>%
# Add significance stars based on posterior probability
mutate(
Star = ifelse(Post.Prob > 95 | Post.Prob < 5, "*", ""),
Estimate = round(Estimate, 3),
Est.Error = round(Est.Error, 3),
CI.Lower = round(CI.Lower, 3),
CI.Upper = round(CI.Upper, 3),
Post.Prob = round(Post.Prob, 2)
) %>%
# Select and arrange the final columns
select(Comparison, Estimate, Est.Error, CI.Lower, CI.Upper, Post.Prob, Star) %>%
arrange(Comparison)diff_1_2: Estimate = -0.058 [-0.096, -0.021], Posterior Probability = 99.9%; There is strong evidence that thef0_slopePostCatvalue increases frompercProm1topercProm2. The negative estimate, combined with a high posterior probability and a credible interval that excludes zero, supports this finding.diff_1_3: Estimate = 0.083 [0.036, 0.128], Posterior Probability = 99.9%; There is strong evidence thatf0_slopePostCatis lower atpercProm3than atpercProm1. The positive estimate, high posterior probability, and credible interval entirely above zero indicate a reliable increase.diff_2_3: Estimate = 0.141 [0.095, 0.186], Posterior Probability = 100%; There is strong evidence thatf0_slopePostCatdecreases frompercProm2topercProm3. The positive estimate and a credible interval that does not include zero affirm this upward trend.
Overall Implications:
Trend: The findings suggest a non-linear trend in
f0_slopePostCatacross prominence levels, with a increase frompercProm1topercProm2followed by a consistent decrease frompercProm1(andpercProm2) topercProm3.Interpretation: The posterior probabilities and credible intervals indicate strong support for these changes across the prominence levels, suggesting that
f0_slopePostCatvaries significantly in response to perceived prominence, with the largest decrease observed betweenpercProm2andpercProm3.
4.1.3.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
data_prepost_cat %>%
data_grid(percProm = unique(percProm),
gender_s = 0.1563) %>% # Set to average of gender
add_epred_draws(mdl_f0_slopePostCat, re_formula = NA) %>%
mutate(
percProm_num = as.numeric(as.character(percProm)),
percProm = factor(percProm, levels = c("1", "2", "3"))
) %>%
ggplot(aes(x = percProm_num, y = .epred, color = percProm)) +
geom_violin(aes(group = percProm_num, fill = percProm), alpha = 0.8) +
scale_color_manual(values = colorBlindBlack8) +
scale_fill_manual(values = colorBlindBlack8) +
stat_summary(fun = median, geom = "point", color = colorBlindBlack8[7], size = 2) +
geom_smooth(
aes(group = 1),
method = "gam",
formula = y ~ s(x, bs = "cs", k = 3),
color = colorBlindBlack8[7],
se = FALSE
) +
scale_x_continuous(
breaks = 0:3,
labels = as.character(0:3)
) +
#ylim(1, 5) +
labs(
x = "Perceived prominence level",
y = "Posterior f0 slope (post-tonic)"
) +
theme_minimal()
post_f0_slopePostCat <-
data_prepost_cat %>%
data_grid(
percProm = unique(percProm),
gender_s = 0.1563 # Set gender_s to zero to average over gender
) %>%
add_epred_draws(mdl_f0_slopePostCat, re_formula = NA) %>%
mutate(
percProm = factor(percProm, levels = c("1", "2", "3"))
) %>%
ggplot(aes(y = percProm, x = .epred, fill = percProm)) +
ggdist::stat_halfeye(
adjust = 0.5,
width = 0.6,
.width = c(0.66, 0.95),
justification = -0.1,
point_colour = NA
) +
geom_boxplot(
width = 0.15,
outlier.shape = NA,
alpha = 0.5,
position = position_nudge(y = 0.2)
) +
stat_summary(
fun = median,
geom = "point",
color = "red",
size = 2,
position = position_nudge(y = 0.2)
) +
scale_fill_manual(values = colorBlindBlack8) +
coord_flip() +
theme_minimal() +
theme(text = element_text(size = 18)) +
#xlim(1, 3.5) +
labs(
y = "Perceived prominence level",
x = "Posterior f0 slope (post-tonic)",
fill = "Perceived\nprominence"
)
post_f0_slopePostCat;
ggsave(plot = post_f0_slopePostCat, filename = paste0(plots, "posterior_f0_slopePostCat.pdf"),
width = 8, height = 6);
ggsave(plot = post_f0_slopePostCat, filename = paste0(plots, "posterior_f0_slopePostCat.jpg"),
width = 8, height = 6);
ggsave(plot = post_f0_slopePostCat, filename = paste0(plots, "posterior_f0_slopePostCat.tif"),
width = 8, height = 6, compression="lzw", dpi=600);4.1.3.4 Group-level effects
First, get a general overview.
as.data.frame(VarCorr(mdl_f0_slopePostCat)$participant$sd) %>%
rownames_to_column(var = "Random Effect") %>%
select(`Random Effect`, Estimate, Est.Error, Q2.5, Q97.5)percProm1: The estimated standard deviation of random effects at
percProm1is 0.027 (95% CrI: 0.001 to 0.063), suggesting limited variability among participants’ responses at this level. Although small, the credible interval indicates some uncertainty, allowing for variability in individual responses.percProm2: The estimated standard deviation at
percProm2is 0.084 (95% CrI: 0.062 to 0.113), indicating moderate variability among participants. The narrower credible interval compared topercProm1suggests that the variation in response is more consistently observed at this level.percProm3: The estimated standard deviation at
percProm3is 0.088 (95% CrI: 0.046 to 0.138), indicating moderate variability. The credible interval highlights that participants show more individualized responses at this level, which could be due to different interpretations of prominence.
Overall Implications:
Trend: Participant variability in
f0_slopePostCattends to increase as the prominence level rises frompercProm1topercProm3.Interpretation: These results suggest that responses at higher prominence levels (
percProm2andpercProm3) exhibit greater participant-specific variability, possibly reflecting more nuanced individual differences in response to perceived prominence at these levels.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
# Extract samples for each participant x percProm
data_prepost_cat %>%
data_grid(
participant = unique(participant),
percProm = unique(percProm),
gender_s = 0.1563 # Average over gender
) %>%
add_epred_draws(mdl_f0_slopePostCat) %>%
median_qi()# Plot the draws for all participants x percProm levels
data_prepost_cat %>%
# Start by selecting unique participants and their gender_s
select(participant, gender_s) %>%
distinct() %>%
# Back-transform gender_s to gender labels
mutate(
gender = if_else(gender_s == -0.5, "Male", "Female")
) %>%
# Combine with percProm levels
crossing(
percProm = unique(data_prepost_cat$percProm)
) %>%
# Generate posterior predictions
add_epred_draws(mdl_f0_slopePostCat) %>%
# Summarize predictions with .66 and .95 intervals
median_qi(.width = c(.66, .95)) %>%
ggplot(aes(x = .epred, y = participant, color = percProm, shape = gender)) +
# Add the thick line for 66% credible interval
geom_linerange(aes(xmin = .lower, xmax = .upper, group = .width),
data = ~ filter(., .width == 0.66), # Use 66% credible interval
size = 1,
alpha = 0.8,
position = position_dodge2(width = 0.7, padding = 0.5)) +
# Add the thin line for 95% credible interval
geom_linerange(aes(xmin = .lower, xmax = .upper, group = .width),
data = ~ filter(., .width == 0.95), # Use 95% credible interval
size = 0.5,
alpha = 0.8,
position = position_dodge2(width = 0.7, padding = 0.5)) +
# Add points with shapes
geom_point(position = position_dodge(width = 0.7), size = 2.5) +
scale_color_manual(values = colorBlindBlack8) +
labs(
x = "Posterior f0 slope (post-tonic)",
y = "Participant",
color = "Perceived prominence",
shape = "Gender"
) +
theme_minimal() +
facet_wrap(~ participant, ncol = 3, scales = "free_y") +
theme(
strip.text = element_blank(),
panel.spacing = unit(0, "lines")
)
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
# We focus on 95% CrI, but this can be adapted in the width parameter
mdl_f0_slopePostCat %>%
spread_draws(r_participant[participant,percProm]) %>%
median_qi(r_participant, .width = c(.95))# In log without Intercept
mdl_f0_slopePostCat %>%
spread_draws(r_participant[participant,percProm]) %>%
median_qi(r_participant, .width = c(.95, .8, .5)) %>%
ggplot(aes(y = as.factor(participant), x = r_participant, color = percProm, xmin = .lower, xmax = .upper)) +
geom_pointinterval(
position = "dodge") +
scale_color_manual(
values = colorBlindBlack8) +
labs(
x = "Individual deviations in f0 slope (post-tonic)",
y = "Participant",
color = "Perceived prominence") +
theme_minimal() +
facet_wrap(~ participant, ncol = 3, scales = "free_y") +
theme(
strip.text = element_blank(),
panel.spacing = unit(0, "lines")
)
4.1.4 Frequency modulation median (post-tonic)
4.1.4.1 Priors
4.1.4.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1697246 0.03077909 0.1188154 0.2373562An R² of approximately 0.17 suggests a modest fit,
indicating that the model explains about 17% of the
variance in the dependent variable fmDep_medianPostCat.
This means that just under one-fifth of the variability in
fmDep_medianPostCat is accounted for by the predictors. The
95% credible interval, which ranges from 11.9% to
23.7%, reflects some uncertainty around the exact R² value but
still suggests a modest level of explanatory power for the model.
4.1.4.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: fmDep_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 1568)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 1.73 0.22 1.36 2.22 1.00 4168
## sd(percProm2) 1.45 0.16 1.18 1.80 1.00 3678
## sd(percProm3) 1.46 0.21 1.12 1.93 1.00 5291
## cor(percProm1,percProm2) 0.91 0.04 0.81 0.97 1.00 4828
## cor(percProm1,percProm3) 0.96 0.03 0.87 1.00 1.00 5600
## cor(percProm2,percProm3) 0.96 0.03 0.89 0.99 1.00 9874
## Tail_ESS
## sd(percProm1) 7223
## sd(percProm2) 6821
## sd(percProm3) 8810
## cor(percProm1,percProm2) 7980
## cor(percProm1,percProm3) 7270
## cor(percProm2,percProm3) 12723
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 0.17 0.16 0.00 0.58 1.00 11132 9714
## percProm2 0.06 0.06 0.00 0.22 1.00 17647 9086
## percProm3 0.19 0.16 0.01 0.61 1.00 15008 9894
## gender_s 0.13 0.12 0.00 0.45 1.00 12949 9816
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 1.15 0.02 1.10 1.19 1.00 27409 11229
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 1.185990 1.171717 1.004751 1.783663
## percProm2 1.060646 1.059923 1.001607 1.242110
## percProm3 1.204865 1.179075 1.005220 1.832147
## gender_s 1.135661 1.129941 1.003480 1.565454It shows a decrease from 1 to 2 and then increase from 2 to 3. 1 and 3 are the same.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.0007 0.1037 0.2250 0.4453 0.4770 6.1503 6784.1.4.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 1.14 1 1.63
## 2 1.05 1 1.20
## 3 1.16 1 1.67
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 0.0794 -0.198 0.631
## percProm1 - percProm3 -0.0130 -0.683 0.615
## percProm2 - percProm3 -0.0995 -0.665 0.195
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.4.3.2 Hypothesis testing
diff_1_2: Estimate = 0.112 [-0.126, 0.532], Posterior Probability = 75.4%; There is moderate evidence suggesting thatfmDep_medianPostCatatpercProm2may be lower than atpercProm1. However, the credible interval includes zero, indicating that this difference is not reliable.diff_1_3: Estimate = -0.016 [-0.474, 0.455], Posterior Probability = 52.9%; There is no strong evidence for a difference betweenpercProm1andpercProm3. The posterior probability is near chance, and the wide credible interval includes zero, making the difference inconclusive.diff_2_3: Estimate = -0.127 [-0.552, 0.124], Posterior Probability = 78.2%; There is moderate evidence thatfmDep_medianPostCatatpercProm3might be higher than atpercProm2, but the credible interval includes zero, suggesting that this difference is not robust.
Overall Implications:
- Trend: There is no clear, strong evidence of
reliable differences in
fmDep_medianPostCatbetween thepercPromlevels. While there is moderate evidence for some comparisons, the inclusion of zero in all credible intervals indicates that the results are inconclusive.
4.1.4.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.4.4 Group-level effects
First, get a general overview.
percProm1: The estimated standard deviation is 1.732 (95% CrI: 1.359 to 2.221), suggesting substantial variability in participants’ responses at
percProm1.percProm2: The estimated standard deviation is 1.451 (95% CrI: 1.179 to 1.798), indicating moderate variability at
percProm2, which is slightly lower than atpercProm1.percProm3: The estimated standard deviation is 1.457 (95% CrI: 1.116 to 1.929), showing similar variability to
percProm2, though with a slightly wider credible interval, suggesting some uncertainty in participants’ responses at this level.
These results highlight that there is considerable participant-level
variability in fmDep_medianPostCat across perceived
prominence levels, with percProm1 showing the greatest
variability.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.5 Shannon entropy SD
4.1.5.1 Priors
4.1.5.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2572849 0.01543586 0.2273162 0.2876225An R² of approximately 0.26 suggests a modest fit.
This value indicates that the model explains about
25.7% of the variance in the dependent variable
entropySh_sd. In other words, slightly more than a quarter
of the variability in entropySh_sd is accounted for by the
predictors in the model. The 95% credible interval, ranging from
22.7% to 28.8%, shows some uncertainty around this
estimate, but it provides a stable indication of the model’s explanatory
power.
4.1.5.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: entropySh_sd ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 2245)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 2.74 0.23 2.31 3.23 1.00 987
## sd(percProm2) 2.61 0.21 2.22 3.05 1.01 922
## sd(percProm3) 2.69 0.24 2.25 3.20 1.00 1180
## cor(percProm1,percProm2) 1.00 0.00 0.99 1.00 1.00 4233
## cor(percProm1,percProm3) 1.00 0.00 0.99 1.00 1.00 4642
## cor(percProm2,percProm3) 1.00 0.00 1.00 1.00 1.00 5539
## Tail_ESS
## sd(percProm1) 1896
## sd(percProm2) 1874
## sd(percProm3) 2310
## cor(percProm1,percProm2) 8042
## cor(percProm1,percProm3) 8224
## cor(percProm2,percProm3) 10058
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 0.16 0.14 0.00 0.54 1.00 15015 8945
## percProm2 0.11 0.11 0.00 0.40 1.00 16910 8873
## percProm3 0.20 0.18 0.01 0.66 1.00 12134 7816
## gender_s 0.28 0.24 0.01 0.91 1.00 3024 3170
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.48 0.01 0.47 0.50 1.00 23597 11899
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 1.169202 1.155980 1.004436 1.714580
## percProm2 1.120618 1.113236 1.002985 1.489015
## percProm3 1.222719 1.196302 1.005946 1.940732
## gender_s 1.322735 1.275764 1.008983 2.472945It shows a decrease from 1 to 2 and then an increase from 2 to 3, and 3 is also sligthly higher than 1.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.005668 0.026568 0.041266 0.044946 0.058181 0.179098 14.1.5.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 1.14 1 1.60
## 2 1.11 1 1.42
## 3 1.19 1 1.78
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 0.0278 -0.385 0.573
## percProm1 - percProm3 -0.0318 -0.782 0.592
## percProm2 - percProm3 -0.0633 -0.736 0.387
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.5.3.2 Hypothesis testing
diff_1_2: Estimate = 0.042 [-0.281, 0.44], Posterior Probability = 58.6%; There is no strong evidence for a difference between
percProm1andpercProm2. The credible interval includes zero, and the posterior probability is close to chance, making the difference unreliable.diff_1_3: Estimate = -0.045 [-0.55, 0.407], Posterior Probability = 57.0%; There is weak evidence suggesting that
entropySh_sdatpercProm3is higher than atpercProm1. However, the wide credible interval and posterior probability indicate a lack of reliability for this difference.diff_2_3: Estimate = -0.087 [-0.562, 0.253], Posterior Probability = 65.9%; There is weak evidence that
entropySh_sdatpercProm3is higher than atpercProm2. This difference remains inconclusive, given the wide credible interval and posterior probability below 95%.
Overall Implications:
- Trend: There is no strong or consistent evidence of
significant differences between perceived prominence levels for
entropySh_sd. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and should be interpreted with caution.
4.1.5.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.5.4 Group-level effects
First, get a general overview.
percProm1: The estimated standard deviation is 2.736 (95% CrI: 2.312 to 3.229), indicating considerable variability in participants’ responses at
percProm1.percProm2: The estimated standard deviation is 2.612 (95% CrI: 2.224 to 3.052), suggesting moderate variability at
percProm2, which is slightly lower than atpercProm1.percProm3: The estimated standard deviation is 2.687 (95% CrI: 2.254 to 3.198), showing similar variability to
percProm1, though with a slightly wider credible interval, suggesting some uncertainty in participants’ responses at this level.
These results highlight that there is considerable participant-level
variability in entropySh_sd across perceived prominence
levels, with relatively high variability observed at each level.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.6 Flux median (post-tonic)
4.1.6.1 Priors
4.1.6.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.02975889 0.003809218 0.02285575 0.03776843An R² of approximately 0.03 suggests a very low fit.
This value indicates that the model explains only about 3% of
the variance in the dependent variable
flux_medianPost. In other words, nearly all of the
variability in flux_medianPost is unaccounted for by the
predictors in the model. The 95% credible interval, ranging from
2.3% to 3.8%, confirms the minimal explanatory power of
the model, indicating high uncertainty in the relationship between the
predictors and the response variable.
4.1.6.3 Fixed effects
Let’s check how it looks like.
## Family: zero_inflated_beta
## Links: mu = logit; phi = identity; zi = identity
## Formula: flux_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 1995)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 3.19 0.27 2.70 3.78 1.01 1259
## sd(percProm2) 2.88 0.22 2.48 3.35 1.01 933
## sd(percProm3) 3.02 0.27 2.54 3.61 1.00 1492
## cor(percProm1,percProm2) 1.00 0.00 0.99 1.00 1.00 5065
## cor(percProm1,percProm3) 0.99 0.01 0.98 1.00 1.00 2878
## cor(percProm2,percProm3) 0.99 0.00 0.98 1.00 1.00 6131
## Tail_ESS
## sd(percProm1) 2297
## sd(percProm2) 2050
## sd(percProm3) 2780
## cor(percProm1,percProm2) 8158
## cor(percProm1,percProm3) 6586
## cor(percProm2,percProm3) 10528
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 0.24 0.22 0.01 0.81 1.00 11696 8382
## percProm2 0.10 0.10 0.00 0.37 1.00 15242 8027
## percProm3 0.27 0.25 0.01 0.94 1.00 11329 8841
## gender_s 0.22 0.21 0.01 0.77 1.00 4738 4152
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## phi 18.82 0.80 17.27 20.41 1.00 22417 10691
## zi 0.00 0.00 0.00 0.00 1.00 20690 9102
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 0.2356616 0.2208363 0.006266123 0.8114322
## percProm2 0.1000393 0.1004706 0.002549701 0.3706760
## percProm3 0.2709642 0.2494974 0.007335838 0.9388155
## gender_s 0.2172763 0.2066923 0.006234345 0.7657176It shows a decrease from 1 to 2, and then an increase from 2 to 3. 3 is more or less as high as 1.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.00000 0.00410 0.00838 0.01712 0.01652 0.99998 2514.1.6.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 0.1712 5.01e-05 0.675
## 2 0.0689 1.89e-05 0.302
## 3 0.2002 2.20e-05 0.773
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## Results are given on the logit (not the response) scale.
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 0.0858 -0.287 0.673
## percProm1 - percProm3 -0.0182 -0.725 0.658
## percProm2 - percProm3 -0.1110 -0.784 0.270
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## Results are given on the log odds ratio (not the response) scale.
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.6.3.2 Hypothesis testing
diff_1_2: Estimate = 0.136 [-0.249, 0.734], Posterior Probability = 71.4%; There is weak evidence suggesting that
flux_medianPostatpercProm2is lower than atpercProm1. However, with a credible interval that includes zero and a posterior probability well below 95%, this difference remains inconclusive.diff_1_3: Estimate = -0.035 [-0.763, 0.627], Posterior Probability = 53.5%; There is no strong evidence for a difference between
percProm1andpercProm3, with a posterior probability close to chance.diff_2_3: Estimate = -0.171 [-0.85, 0.232], Posterior Probability = 74.4%; There is weak evidence that
flux_medianPostatpercProm3is higher than atpercProm2, but this result is inconclusive due to the credible interval including zero.
Overall Implications:
- Trend: There is no consistent or strong evidence of
significant differences between perceived prominence levels for
flux_medianPost. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and lack reliability.
4.1.6.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.6.4 Group-level effects
First, get a general overview.
percProm1: The estimated standard deviation is 3.185 (95% CrI: 2.705 to 3.780), indicating considerable variability in participants’ responses at
percProm1.percProm2: The estimated standard deviation is 2.875 (95% CrI: 2.477 to 3.352), suggesting moderate participant-level variability at
percProm2, slightly lower than atpercProm1.percProm3: The estimated standard deviation is 3.016 (95% CrI: 2.541 to 3.611), showing similar variability to
percProm1andpercProm2.
These results highlight substantial participant-level variability in
flux_medianPost across perceived prominence levels,
suggesting that individual differences play a significant role in the
responses at each prominence level.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.7 Amplitude median (post-tonic)
4.1.7.1 Priors
4.1.7.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2911882 0.01956155 0.2538602 0.330177An R² of approximately 0.29 suggests a moderate fit, indicating that
the model explains about 29.1% of the variance in the dependent variable
ampl_medianPost. Thus, nearly a third of the variability in
ampl_medianPost is accounted for by the predictors. The 95%
credible interval (25.4% to 33.0%) indicates moderate certainty around
this explanatory power.
4.1.7.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: ampl_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 1995)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 2.42 0.23 2.01 2.91 1.00 1550
## sd(percProm2) 2.18 0.21 1.81 2.63 1.00 1599
## sd(percProm3) 2.36 0.23 1.96 2.87 1.00 1789
## cor(percProm1,percProm2) 0.98 0.01 0.97 0.99 1.00 3695
## cor(percProm1,percProm3) 0.97 0.01 0.93 0.99 1.00 3999
## cor(percProm2,percProm3) 0.99 0.00 0.98 1.00 1.00 6398
## Tail_ESS
## sd(percProm1) 3003
## sd(percProm2) 3257
## sd(percProm3) 4104
## cor(percProm1,percProm2) 8031
## cor(percProm1,percProm3) 7858
## cor(percProm2,percProm3) 10916
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 0.12 0.12 0.00 0.44 1.00 16433 9810
## percProm2 0.13 0.12 0.00 0.45 1.00 16053 8821
## percProm3 0.16 0.15 0.00 0.54 1.00 18455 10168
## gender_s 0.23 0.20 0.01 0.76 1.00 5563 5200
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.67 0.01 0.65 0.69 1.00 31800 11661
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 1.129096 1.126741 1.003055 1.551368
## percProm2 1.144207 1.127858 1.004314 1.563297
## percProm3 1.173223 1.157395 1.004957 1.718642
## gender_s 1.254358 1.226229 1.006970 2.132875It shows an increase from 1 to 2, and then a decrease from 2 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.00330 0.04785 0.08166 0.10006 0.13731 0.34435 2514.1.7.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 1.10 1 1.46
## 2 1.12 1 1.49
## 3 1.14 1 1.59
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 -0.0118 -0.438 0.448
## percProm1 - percProm3 -0.0278 -0.596 0.440
## percProm2 - percProm3 -0.0157 -0.559 0.429
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.7.3.2 Hypothesis testing
diff_1_2: Estimate = -0.013 [-0.357, 0.331], Posterior Probability = 54.2%; There is no strong evidence for a difference between
percProm1andpercProm2, with the posterior probability close to chance.diff_1_3: Estimate = -0.038 [-0.445, 0.334], Posterior Probability = 58.0%; There is weak evidence suggesting
ampl_medianPostatpercProm3may be slightly lower thanpercProm1, but it remains inconclusive.diff_2_3: Estimate = -0.025 [-0.424, 0.327], Posterior Probability = 54.7%; No strong evidence for a difference between
percProm2andpercProm3, with the posterior probability close to chance.
Overall Implications: There is no strong evidence of
consistent differences in ampl_medianPost across perceived
prominence levels, indicating that the predictor levels do not
significantly impact ampl_medianPost.
4.1.7.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.7.4 Group-level effects
First, get a general overview.
percProm1: Estimated SD = 2.421 (95% CrI: 2.014 to 2.913)
percProm2: Estimated SD = 2.181 (95% CrI: 1.809 to 2.630)
percProm3: Estimated SD = 2.363 (95% CrI: 1.963 to 2.875)
Substantial variability is evident across levels, suggesting individual differences in responses to each prominence level.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.8 Spectral centroid median
4.1.8.1 Priors
4.1.8.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2441985 0.01455366 0.2157455 0.2728663An R² of approximately 0.24 suggests a modest fit, with the model
explaining around 24.4% of the variance in
specCentroid_median. The 95% credible interval (21.6% to
27.3%) suggests some certainty in this explanatory power, though room
for improvement remains.
4.1.8.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: specCentroid_median ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 2245)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 0.08 0.01 0.06 0.10 1.00 5503
## sd(percProm2) 0.05 0.01 0.04 0.07 1.00 5568
## sd(percProm3) 0.05 0.01 0.04 0.08 1.00 8711
## cor(percProm1,percProm2) 0.73 0.10 0.49 0.89 1.00 5588
## cor(percProm1,percProm3) 0.57 0.17 0.20 0.85 1.00 7142
## cor(percProm2,percProm3) 0.88 0.09 0.66 0.99 1.00 8314
## Tail_ESS
## sd(percProm1) 8775
## sd(percProm2) 8328
## sd(percProm3) 11366
## cor(percProm1,percProm2) 9077
## cor(percProm1,percProm3) 10759
## cor(percProm2,percProm3) 10529
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 5.76 0.01 5.74 5.79 1.00 3827 5694
## percProm2 5.75 0.01 5.73 5.77 1.00 4177 7493
## percProm3 5.72 0.01 5.70 5.74 1.00 5959 8938
## gender_s 0.01 0.01 0.00 0.03 1.00 10159 9555
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.10 0.00 0.09 0.10 1.00 26006 12346
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 318.731797 1.014300 309.90812 327.947156
## percProm2 315.211758 1.009359 309.43561 321.095564
## percProm3 304.976537 1.011357 298.23076 311.820390
## gender_s 1.008489 1.007830 1.00025 1.029005It shows a decrease from 1 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 217.1 291.6 314.2 316.7 340.7 421.8 14.1.8.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 319 310 328
## 2 315 310 321
## 3 305 298 312
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 3.53 -3.17 10.1
## percProm1 - percProm3 13.74 4.87 21.8
## percProm2 - percProm3 10.24 5.37 15.2
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.8.3.2 Hypothesis testing
diff_1_2: Estimate = 3.54 [-3.07, 10.3], Posterior Probability = 85.7%; Weak evidence that
specCentroid_medianatpercProm2is lower than atpercProm1, but this difference is not conclusive.diff_1_3: Estimate = 13.8 [5.38, 22.4], Posterior Probability = 99.9%; Strong evidence that
specCentroid_medianis lower atpercProm3than atpercProm1.diff_2_3: Estimate = 10.2 [5.26, 15.1], Posterior Probability = 100%; Strong evidence that
specCentroid_medianis lower atpercProm3than atpercProm2.
Overall Implications: A trend is observed, where
specCentroid_median decreases with perceived prominence
levels, particularly between percProm1 and
percProm3.
4.1.8.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.8.4 Group-level effects
First, get a general overview.
percProm1: Estimated SD = 0.078 (95% CrI: 0.058 to 0.103)
percProm2: Estimated SD = 0.052 (95% CrI: 0.04 to 0.068)
percProm3: Estimated SD = 0.054 (95% CrI: 0.038 to 0.075)
Variability remains moderate across levels, suggesting participant responses vary slightly based on perceived prominence levels.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.9 Duration (without silences)
4.1.9.1 Priors
4.1.9.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1999007 0.01466163 0.1713953 0.2290767An R² of approximately 0.20 indicates a modest fit, with the model
explaining about 20.0% of the variance in
duration_noSilence. The credible interval (17.1% to 22.9%)
indicates moderate certainty around this explanatory power.
4.1.9.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: duration_noSilence ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 2245)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 0.16 0.03 0.11 0.23 1.00 7334
## sd(percProm2) 0.14 0.02 0.11 0.19 1.00 7138
## sd(percProm3) 0.18 0.03 0.12 0.25 1.00 8836
## cor(percProm1,percProm2) 0.66 0.15 0.32 0.89 1.00 4682
## cor(percProm1,percProm3) 0.63 0.19 0.19 0.93 1.00 5587
## cor(percProm2,percProm3) 0.79 0.12 0.48 0.97 1.00 7956
## Tail_ESS
## sd(percProm1) 10567
## sd(percProm2) 11384
## sd(percProm3) 11596
## cor(percProm1,percProm2) 7595
## cor(percProm1,percProm3) 7814
## cor(percProm2,percProm3) 9856
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 5.05 0.03 4.98 5.11 1.00 6212 9625
## percProm2 5.15 0.03 5.10 5.20 1.00 6246 9761
## percProm3 5.29 0.04 5.21 5.37 1.00 7375 11658
## gender_s 0.08 0.05 0.01 0.18 1.00 6689 7514
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.32 0.00 0.31 0.33 1.00 29897 11437
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 155.877398 1.034254 145.868565 166.436640
## percProm2 172.523441 1.027185 163.650036 181.817854
## percProm3 198.881462 1.040217 183.619247 214.274873
## gender_s 1.083926 1.046648 1.006463 1.194102It shows a stable increase from 1 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 55.0 135.0 175.0 187.7 225.0 685.0 14.1.9.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 156 146 166
## 2 173 164 182
## 3 199 184 214
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 -16.6 -25.7 -6.98
## percProm1 - percProm3 -43.2 -57.9 -29.43
## percProm2 - percProm3 -26.5 -38.4 -14.11
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.9.3.2 Hypothesis testing
diff_1_2: Estimate = -16.6 [-26.0, -7.29], Posterior Probability = 99.9%; Strong evidence that
duration_noSilenceincreases frompercProm1topercProm2.diff_1_3: Estimate = -43.1 [-57.6, -29.0], Posterior Probability = 100%; Strong evidence that
duration_noSilenceincreases frompercProm1topercProm3.diff_2_3: Estimate = -26.4 [-38.7, -14.4], Posterior Probability = 100%; Strong evidence that
duration_noSilenceincreases frompercProm2topercProm3.
Overall Implications:
duration_noSilence tends to increase as perceived
prominence levels increase, with consistent evidence supporting this
trend.
4.1.9.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.9.4 Group-level effects
First, get a general overview.
percProm1: Estimated SD = 0.162 (95% CrI: 0.113 to 0.226)
percProm2: Estimated SD = 0.143 (95% CrI: 0.109 to 0.188)
percProm3: Estimated SD = 0.177 (95% CrI: 0.117 to 0.251)
Variability shows modest participant-level differences across perceived prominence levels.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.10 Duration (pretonic)
4.1.10.1 Priors
4.1.10.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1239293 0.0153715 0.09510713 0.1551957An R² of approximately 0.12 indicates a relatively low fit,
suggesting the model explains around 12.4% of the variance in
durationPre. The 95% credible interval (9.5% to 15.5%)
indicates some uncertainty in the explanatory power of the model.
4.1.10.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: durationPre ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 2063)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 0.13 0.03 0.08 0.20 1.00 5841
## sd(percProm2) 0.10 0.02 0.07 0.13 1.00 7318
## sd(percProm3) 0.22 0.04 0.15 0.30 1.00 7764
## cor(percProm1,percProm2) 0.72 0.17 0.30 0.97 1.00 2783
## cor(percProm1,percProm3) 0.50 0.25 -0.04 0.90 1.00 2691
## cor(percProm2,percProm3) 0.83 0.12 0.53 0.98 1.00 7378
## Tail_ESS
## sd(percProm1) 9488
## sd(percProm2) 10653
## sd(percProm3) 10926
## cor(percProm1,percProm2) 5667
## cor(percProm1,percProm3) 4202
## cor(percProm2,percProm3) 11626
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 4.91 0.03 4.84 4.97 1.00 6898 9925
## percProm2 4.98 0.02 4.94 5.02 1.00 6342 10154
## percProm3 5.13 0.05 5.04 5.22 1.00 7834 10167
## gender_s 0.06 0.03 0.01 0.14 1.00 6377 5080
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.38 0.01 0.37 0.39 1.00 23572 11765
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 135.038298 1.033495 126.611126 144.033299
## percProm2 146.197507 1.020698 140.468609 152.149767
## percProm3 168.879341 1.047879 153.745826 184.676868
## gender_s 1.066978 1.035397 1.006147 1.147767It shows a stable increase from 1 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 31.0 116.0 146.0 161.7 191.0 573.0 1834.1.10.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 135 126 144
## 2 146 141 152
## 3 169 154 185
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 -11.1 -18.9 -2.89
## percProm1 - percProm3 -34.0 -49.1 -18.65
## percProm2 - percProm3 -22.8 -35.9 -9.95
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.10.3.2 Hypothesis testing
diff_1_2: Estimate = -11.1 [-19.2, -3.09], Posterior Probability = 99.6%; Strong evidence that
durationPreincreases frompercProm1topercProm2.diff_1_3: Estimate = -34.0 [-49.4, -18.9], Posterior Probability = 100%; Strong evidence that
durationPreincreases frompercProm1topercProm3.diff_2_3: Estimate = -22.8 [-36.0, -10.0], Posterior Probability = 100%; Strong evidence that
durationPreincreases frompercProm2topercProm3.
Overall Implications: A consistent pattern is
observed where durationPre increases as perceived
prominence levels increase, with high confidence in these
differences.
4.1.10.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.10.4 Group-level effects
First, get a general overview.
percProm1: Estimated SD = 0.135 (95% CrI: 0.079 to 0.205)
percProm2: Estimated SD = 0.098 (95% CrI: 0.070 to 0.134)
percProm3: Estimated SD = 0.219 (95% CrI: 0.149 to 0.305)
The variability is somewhat pronounced at percProm3,
suggesting that participants vary in responses to perceived
prominence.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.1.11 Amplitude SD
4.1.11.1 Priors
4.1.11.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2717001 0.01602145 0.2411063 0.3034417An R² of approximately 0.27 suggests a moderate fit, indicating the
model explains about 27.2% of the variance in ampl_sd. The
credible interval (24.1% to 30.3%) suggests moderate certainty around
the explanatory power.
4.1.11.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: ampl_sd ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_cat (Number of observations: 2245)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm1) 2.92 0.25 2.48 3.46 1.00 970
## sd(percProm2) 2.71 0.23 2.30 3.20 1.00 966
## sd(percProm3) 2.71 0.23 2.29 3.21 1.00 1033
## cor(percProm1,percProm2) 0.99 0.00 0.99 1.00 1.00 3734
## cor(percProm1,percProm3) 0.99 0.01 0.97 1.00 1.00 4060
## cor(percProm2,percProm3) 1.00 0.00 0.99 1.00 1.00 7442
## Tail_ESS
## sd(percProm1) 2013
## sd(percProm2) 1928
## sd(percProm3) 2173
## cor(percProm1,percProm2) 8246
## cor(percProm1,percProm3) 7777
## cor(percProm2,percProm3) 12335
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm1 0.15 0.14 0.00 0.52 1.00 13096 9095
## percProm2 0.15 0.13 0.00 0.50 1.00 12439 8293
## percProm3 0.15 0.14 0.00 0.52 1.00 14481 8110
## gender_s 0.24 0.22 0.01 0.83 1.00 3635 3687
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.64 0.01 0.62 0.66 1.00 24073 12312
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm1 1.161439 1.153220 1.003742 1.688306
## percProm2 1.162384 1.142325 1.004364 1.641270
## percProm3 1.165793 1.151592 1.004306 1.686098
## gender_s 1.275089 1.251957 1.006832 2.293520It shows that the DV stays more or less the same.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.002609 0.019816 0.036262 0.042747 0.059618 0.146670 14.1.11.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 1 1.13 1 1.56
## 2 1.14 1 1.54
## 3 1.14 1 1.57
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm1 - percProm2 -0.004149 -0.516 0.522
## percProm1 - percProm3 -0.004562 -0.576 0.565
## percProm2 - percProm3 -0.000474 -0.527 0.510
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.1.11.3.2 Hypothesis testing
diff_1_2: Estimate = -0.001 [-0.377, 0.399], Posterior Probability = 51.3%; No strong evidence for a difference between
percProm1andpercProm2.diff_1_3: Estimate = -0.004 [-0.415, 0.423], Posterior Probability = 51.3%; No conclusive evidence for a difference between
percProm1andpercProm3.diff_2_3: Estimate = -0.003 [-0.39, 0.375], Posterior Probability = 50.2%; No strong evidence for a difference between
percProm2andpercProm3.
Overall Implications: There is no strong evidence of
significant differences in ampl_sd across perceived
prominence levels, suggesting a stable response across conditions.
4.1.11.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.1.11.4 Group-level effects
First, get a general overview.
percProm1: Estimated SD = 2.924 (95% CrI: 2.484 to 3.456)
percProm2: Estimated SD = 2.711 (95% CrI: 2.296 to 3.203)
percProm3: Estimated SD = 2.710 (95% CrI: 2.294 to 3.212)
Participant-level variability remains substantial across conditions, indicating differences in individual responses.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2 German
Let’s check the correlations between acoustic features.
Let’s set contrasts for comparisons. (Only for intercept models!)
What is the mean for gender.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -0.50000 -0.50000 0.50000 0.09727 0.50000 0.50000We go one by one the ten features from top to bottom.
4.2.1 F0 slope
4.2.1.1 Priors
4.2.1.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.1593904 0.01067803 0.1386337 0.1803713An R² of approximately 0.16 suggests a weak fit.
This means the model explains only about 15.9% of the variance in the
dependent variable f0_slopeGer, indicating that the
majority of the variability in f0_slopeGer is not accounted for by the
predictors in the model. The 95% credible interval, ranging from
13.9% to 18.0%, highlights the uncertainty but confirms
that the model’s explanatory power is relatively low.
4.2.1.3 Fixed effects
Let’s check how it looks like.
## Family: student
## Links: mu = identity; sigma = identity; nu = identity
## Formula: f0_slope ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2246)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 0.11 0.06 0.02 0.27 1.00 2996
## sd(percProm1) 0.05 0.01 0.03 0.07 1.00 9455
## sd(percProm2) 0.09 0.01 0.07 0.11 1.00 8270
## sd(percProm3) 0.11 0.02 0.07 0.15 1.00 10020
## cor(percProm0,percProm1) 0.11 0.35 -0.60 0.75 1.00 2399
## cor(percProm0,percProm2) -0.02 0.32 -0.68 0.55 1.00 1757
## cor(percProm1,percProm2) 0.59 0.18 0.18 0.87 1.00 5772
## cor(percProm0,percProm3) -0.34 0.31 -0.86 0.31 1.00 2591
## cor(percProm1,percProm3) 0.17 0.25 -0.32 0.64 1.00 7457
## cor(percProm2,percProm3) 0.58 0.16 0.23 0.84 1.00 11866
## Tail_ESS
## sd(percProm0) 5033
## sd(percProm1) 11365
## sd(percProm2) 11868
## sd(percProm3) 12232
## cor(percProm0,percProm1) 3985
## cor(percProm0,percProm2) 2417
## cor(percProm1,percProm2) 9302
## cor(percProm0,percProm3) 5257
## cor(percProm1,percProm3) 10483
## cor(percProm2,percProm3) 13406
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 -0.06 0.05 -0.15 0.04 1.00 8742 7313
## percProm1 -0.00 0.01 -0.02 0.02 1.00 11442 12028
## percProm2 0.12 0.02 0.09 0.15 1.00 9477 10920
## percProm3 0.19 0.02 0.14 0.23 1.00 12440 11779
## gender_s -0.03 0.02 -0.07 0.01 1.00 10242 11207
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.14 0.00 0.13 0.15 1.00 20735 12679
## nu 3.05 0.23 2.63 3.54 1.00 22918 12406
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 -0.0563990095 0.04852796 -0.14768697 0.04369150
## percProm1 -0.0001312484 0.01121977 -0.02217870 0.02227687
## percProm2 0.1232162382 0.01573376 0.09214329 0.15437697
## percProm3 0.1874144619 0.02184852 0.14323515 0.23029715
## gender_s -0.0273822851 0.01996702 -0.06684604 0.01188390It shows a gradual increase in f0_slope from percProm 0, through 1, 2, up to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## -2.86180 -0.04065 0.07234 0.09222 0.22678 2.54937 98Mean is the same, the posterior averages including RE correspond to raw values.
4.2.1.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 -0.057886 -0.1518 0.0383
## 1 -0.000132 -0.0224 0.0218
## 2 0.123204 0.0913 0.1535
## 3 0.187518 0.1413 0.2280
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 -0.0574 -0.152 0.0411
## percProm0 - percProm2 -0.1806 -0.280 -0.0792
## percProm0 - percProm3 -0.2450 -0.351 -0.1343
## percProm1 - percProm2 -0.1233 -0.152 -0.0948
## percProm1 - percProm3 -0.1877 -0.234 -0.1422
## percProm2 - percProm3 -0.0642 -0.104 -0.0250
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95If the comparison includes 0, the contrast is not reliably different. For example:
percProm0 - percProm1: The estimated difference in
f0_slope between percProm0 and
percProm1 is -0.0574. The HPD interval is
(-0.152, 0.0411), which includes zero, meaning this
difference is not credibly different from zero (i.e.,
no significant difference between these levels).
percProm0 - percProm2: The estimated difference
between percProm0 and percProm2 is
-0.1806, and the HPD interval is (-0.280,
-0.0792). Since this interval does not include
zero, the difference is credibly different from
zero, indicating a significant difference in
f0_slope between these two levels.
4.2.1.3.2 Hypothesis testing
In our analysis, we are interested in comparing all four levels of
perceived prominence (percProm) with each other to
determine if there are significant differences in f0_slope
across these levels. Since we are using a no-intercept
model (i.e.,
f0_slope ~ 0 + percProm + gender + (0 + percProm | participant)),
the fixed effects coefficients directly represent the mean
f0_slope for each percProm level.
We choose to use our own custom pipeline instead of the
hypothesis() function for the following reasons:
Direct Access to Posterior Samples: By extracting the posterior samples of the fixed effects coefficients, we can directly compute the differences between
percPromlevels. This approach allows us to utilize the full posterior distribution for each level in our comparisons.Flexible and Comprehensive Comparisons: Our pipeline enables us to compute all pairwise comparisons between
percPromlevels in a single, cohesive process. This flexibility is beneficial when dealing with multiple levels and numerous comparisons.Customized Posterior Probability Calculations: We can calculate the posterior probabilities that the differences are greater than or less than zero, providing a clear measure of the strength of evidence for each comparison.
Simplified Interpretation: The results from our pipeline are presented in an easy-to-interpret format, including estimates, standard errors, credible intervals, and significance indicators, similar to the output of
hypothesis()but tailored to our specific model structure.
Using our custom pipeline allows us to:
Compute differences between specific levels of
percPromdirectly from the posterior samples of the fixed effects.Obtain posterior probabilities for these differences, giving us insight into the likelihood that a true difference exists.
Present the results in a clear and interpretable manner, facilitating straightforward comparison and interpretation.
diff_0_1: Estimate = -0.056 [-0.149, 0.045], Posterior Probability = 89.40%; There is moderate evidence that thef0_slopeatpercProm1is higher than atpercProm0. However, since the credible interval includes zero and the posterior probability is less than 95%, this difference is not considered reliable.diff_0_2: Estimate = -0.180 [-0.278, -0.076], Posterior Probability = 99.60%; Strong evidence that thef0_slopeatpercProm2is higher than atpercProm0. The negative estimate indicates thatpercProm0has a lowerf0_slopecompared topercProm2.diff_0_3: Estimate = -0.244 [-0.350, -0.134], Posterior Probability = 99.90%; Strong evidence that thef0_slopeatpercProm3is higher than atpercProm0.diff_1_2: Estimate = -0.123 [-0.153, -0.095], Posterior Probability = 100%; Strong evidence that thef0_slopeincreases frompercProm1topercProm2.diff_1_3: Estimate = -0.188 [-0.231, -0.141], Posterior Probability = 100%; Strong evidence that thef0_slopeincreases frompercProm1topercProm3.diff_2_3: Estimate = -0.064 [-0.104, -0.024], Posterior Probability = 99.90%; Strong evidence that thef0_slopeincreases frompercProm2topercProm3.
Overall Implications:
Trend: The
f0_slopeincreases aspercPromlevels increase from 0 to 3.Interpretation: Except for the comparison between levels 0 and 1, all other comparisons show strong evidence (posterior probability > 95%) of an increase in
f0_slopewith increasingpercProm. As the perceived prominence (percProm) increases, thef0_slopebecomes more positive (or less negative), indicating a rising pitch slope. This suggests that higher prominence levels are associated with an increase inf0_slope.
4.2.1.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.1.4 Group-level effects
First, get a general overview.
percProm0: The estimated standard deviation of the random intercepts is 0.1095 (95% CrI: 0.0168 to 0.2662), indicating moderate variability in participants’ baseline
f0_slope.percProm1: The estimated standard deviation is 0.0506 (95% CrI: 0.0306 to 0.0737), suggesting some variability in participants’ response to
percProm1.percProm2: The estimated standard deviation is 0.0862 (95% CrI: 0.0656 to 0.1133), indicating some (more that
percProm1) variability in participants’ response topercProm2.percProm3: The estimated standard deviation is 0.1052 (95% CrI: 0.0704 to 0.1492), indicating slightly more variability in response to
percProm3compared topercProm2.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.2 F0 median
4.2.2.1 Priors
4.2.2.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5735031 0.009071667 0.5551615 0.5906223An R² of around 0.57 suggests a moderately strong
fit. This indicates that the model explains approximately 57.4%
of the variance in the dependent variable, meaning more than half of the
variability in pitch_medianGer is accounted for by the
predictors. The 95% credible interval suggests the R² value is likely
between 55.5% and 59.1%, indicating the model’s
performance is stable and provides a relatively good explanation of the
variance.
4.2.2.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: pitch_median ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2156)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 0.29 0.08 0.16 0.48 1.00 7315
## sd(percProm1) 0.09 0.01 0.06 0.12 1.00 6263
## sd(percProm2) 0.07 0.01 0.05 0.10 1.00 6426
## sd(percProm3) 0.11 0.02 0.08 0.15 1.00 9066
## cor(percProm0,percProm1) 0.20 0.28 -0.39 0.69 1.00 1904
## cor(percProm0,percProm2) 0.15 0.25 -0.36 0.61 1.00 2334
## cor(percProm1,percProm2) 0.63 0.14 0.32 0.85 1.00 6746
## cor(percProm0,percProm3) -0.19 0.29 -0.70 0.38 1.00 2741
## cor(percProm1,percProm3) 0.43 0.17 0.06 0.73 1.00 8784
## cor(percProm2,percProm3) 0.72 0.12 0.43 0.91 1.00 9143
## Tail_ESS
## sd(percProm0) 9840
## sd(percProm1) 8964
## sd(percProm2) 9368
## sd(percProm3) 12342
## cor(percProm0,percProm1) 3646
## cor(percProm0,percProm2) 4763
## cor(percProm1,percProm2) 10528
## cor(percProm0,percProm3) 6221
## cor(percProm1,percProm3) 11645
## cor(percProm2,percProm3) 13242
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 5.17 0.09 4.99 5.34 1.00 5311 9008
## percProm1 5.11 0.02 5.08 5.15 1.00 7755 9988
## percProm2 5.17 0.01 5.14 5.20 1.00 8204 10643
## percProm3 5.23 0.02 5.19 5.28 1.00 9812 12084
## gender_s 0.36 0.03 0.31 0.42 1.00 5940 8604
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.15 0.00 0.15 0.16 1.00 27528 11664
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 175.184208 1.091839 146.704657 207.733316
## percProm1 166.258631 1.017176 160.676166 171.803876
## percProm2 175.728060 1.013381 171.248364 180.477361
## percProm3 187.510857 1.021714 179.826897 195.759126
## gender_s 1.439071 1.028551 1.360768 1.520318It shows that pitch median falls from 0 to 1, but raises gradually from 1, through 2 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 90.55 152.38 191.52 188.92 218.30 475.62 1884.2.2.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 178 148 210
## 1 169 163 175
## 2 179 174 183
## 3 191 183 199
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 9.003 -20.0 42.12
## percProm0 - percProm2 -0.571 -29.9 32.00
## percProm0 - percProm3 -12.540 -44.1 20.88
## percProm1 - percProm2 -9.654 -14.6 -4.82
## percProm1 - percProm3 -21.581 -29.7 -13.79
## percProm2 - percProm3 -11.960 -18.6 -5.82
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.2.3.2 Hypothesis testing
diff_0_1: Estimate = 0.052 [-0.124, 0.226], Posterior Probability = 73.90%; There is weak evidence that thepitch_medianatpercProm1is higher than atpercProm0. However, the posterior probability is low, and the credible interval includes zero, suggesting that this difference is not reliable.diff_0_2: Estimate = -0.003 [-0.179, 0.168], Posterior Probability = 51.40%; No evidence for a difference betweenpercProm0andpercProm2. The estimate is nearly zero, and the credible interval includes zero, making this difference insignificant.diff_0_3: Estimate = -0.068 [-0.251, 0.114], Posterior Probability = 77.90%; There is weak evidence thatpitch_medianatpercProm3is higher than atpercProm0. However, since the credible interval includes zero and the posterior probability is less than 95%, this difference is not considered reliable.diff_1_2: Estimate = -0.055 [-0.084, -0.027], Posterior Probability = 100%; Strong evidence that thepitch_medianincreases frompercProm1topercProm2. The negative estimate and the credible interval fully below zero support this.diff_1_3: Estimate = -0.120 [-0.164, -0.077], Posterior Probability = 100%; Strong evidence that thepitch_medianincreases frompercProm1topercProm3.diff_2_3: Estimate = -0.065 [-0.099, -0.032], Posterior Probability = 100%; Strong evidence that thepitch_medianincreases frompercProm2topercProm3.
Overall Implications:
Trend: The
pitch_mediangenerally increases aspercPromlevels increase from 1 to 3. However, there is weak or no evidence of a difference betweenpercProm0and the other levels.Interpretation: Strong evidence (posterior probability ≥ 95%) is present for increases in
pitch_medianbetween levelspercProm1,percProm2, andpercProm3. However, comparisons involvingpercProm0do not show reliable differences.
4.2.2.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.2.4 Group-level effects
First, get a general overview.
percProm0: The estimated standard deviation of the random effects for
percProm0is 0.2853 (95% CrI: 0.1642 to 0.4785), indicating substantial variability in participants’ baselinepitch_median.percProm1: The estimated standard deviation is 0.0892 (95% CrI: 0.0649 to 0.1211), suggesting relatively low variability in participants’ response to
percProm1.percProm2: The estimated standard deviation is 0.0716 (95% CrI: 0.0541 to 0.0951), indicating even lower variability in participants’ response to
percProm2.percProm3: The estimated standard deviation is 0.1068 (95% CrI: 0.0771 to 0.1451), indicating moderate variability in participants’ response to
percProm3, slightly higher than forpercProm2.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.3 Amplitude SD
4.2.3.1 Priors
4.2.3.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5280124 0.01575281 0.4959289 0.5576492An R² of around 0.53 suggests a moderate fit. The
model explains approximately 52.8% of the variance in the dependent
variable, meaning just over half of the variability in
ampl_sdGer is accounted for by the predictors. The 95%
credible interval shows some uncertainty about the exact R² value, but
it is likely between 49.6% and 55.8%. This range
indicates that the model’s ability to explain the variance is relatively
stable.
4.2.3.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: ampl_sd ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2344)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 3.65 0.27 3.17 4.23 1.00 1802
## sd(percProm1) 3.12 0.21 2.74 3.56 1.00 1625
## sd(percProm2) 2.99 0.21 2.61 3.42 1.00 1795
## sd(percProm3) 2.88 0.23 2.48 3.37 1.00 2245
## cor(percProm0,percProm1) 0.98 0.01 0.96 1.00 1.00 3080
## cor(percProm0,percProm2) 0.98 0.01 0.95 0.99 1.00 2379
## cor(percProm1,percProm2) 1.00 0.00 1.00 1.00 1.00 6207
## cor(percProm0,percProm3) 0.97 0.01 0.94 0.99 1.00 2268
## cor(percProm1,percProm3) 0.99 0.00 0.98 1.00 1.00 5932
## cor(percProm2,percProm3) 1.00 0.00 0.99 1.00 1.00 6950
## Tail_ESS
## sd(percProm0) 3236
## sd(percProm1) 2824
## sd(percProm2) 3295
## sd(percProm3) 4186
## cor(percProm0,percProm1) 5671
## cor(percProm0,percProm2) 4535
## cor(percProm1,percProm2) 10932
## cor(percProm0,percProm3) 4524
## cor(percProm1,percProm3) 10486
## cor(percProm2,percProm3) 10815
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 0.24 0.22 0.01 0.80 1.00 12271 8363
## percProm1 0.11 0.11 0.00 0.40 1.00 16235 8768
## percProm2 0.19 0.15 0.01 0.55 1.00 12832 8215
## percProm3 0.28 0.23 0.01 0.84 1.00 12511 9182
## gender_s 0.24 0.22 0.01 0.81 1.00 7253 6326
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.49 0.01 0.47 0.50 1.00 25941 11257
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 1.265566 1.244857 1.005939 2.230485
## percProm1 1.121461 1.114886 1.002993 1.498118
## percProm2 1.203757 1.160954 1.006890 1.737447
## percProm3 1.323594 1.253977 1.009336 2.305958
## gender_s 1.273367 1.240224 1.007432 2.244647It shows that ampl_sd falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0006497 0.0075126 0.0123106 0.0156947 0.0202742 0.08210184.2.3.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 1.20 1 2.00
## 1 1.10 1 1.41
## 2 1.18 1 1.64
## 3 1.27 1 2.09
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 0.0839 -0.460 0.971
## percProm0 - percProm2 0.0245 -0.683 0.917
## percProm0 - percProm3 -0.0501 -1.130 0.974
## percProm1 - percProm2 -0.0617 -0.587 0.330
## percProm1 - percProm3 -0.1515 -1.109 0.354
## percProm2 - percProm3 -0.0805 -1.008 0.494
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.3.3.2 Hypothesis testing
diff_0_1: Estimate = 0.121 [-0.277, 0.712], Posterior Probability = 68.10%; There is weak evidence that theampl_sdatpercProm1is higher than atpercProm0. However, the posterior probability is relatively low, and the credible interval includes zero, suggesting that this difference is not reliable.diff_0_2: Estimate = 0.050 [-0.419, 0.666], Posterior Probability = 54.50%; There is no strong evidence for a difference betweenpercProm0andpercProm2. The credible interval spans zero, and the posterior probability is low.diff_0_3: Estimate = -0.045 [-0.686, 0.614], Posterior Probability = 57.00%; There is no clear evidence that theampl_sdatpercProm0differs from that atpercProm3. The posterior probability is too low to support a reliable conclusion.diff_1_2: Estimate = -0.071 [-0.441, 0.252], Posterior Probability = 66.20%; There is moderate but weak evidence suggesting that theampl_sdatpercProm1is lower than atpercProm2, but the credible interval includes zero.diff_1_3: Estimate = -0.166 [-0.738, 0.243], Posterior Probability = 74.90%; There is moderate evidence that theampl_sdatpercProm1is lower than atpercProm3, but this difference is not strong enough to be considered conclusive.diff_2_3: Estimate = -0.095 [-0.639, 0.342], Posterior Probability = 63.70%; There is no clear evidence for a difference betweenpercProm2andpercProm3, as the credible interval includes zero and the posterior probability is moderate.
Overall Implications:
Trend: There is no strong evidence of consistent differences in
ampl_sdacrosspercPromlevels. Most comparisons show weak or inconclusive results, with posterior probabilities below 95%, and credible intervals that include zero.Interpretation: The comparisons do not provide strong support for significant variability in
ampl_sdacross the perceived prominence levels, suggesting that changes inpercPrommay not have a substantial impact onampl_sd.
4.2.3.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.3.4 Group-level effects
First, get a general overview.
percProm0: The estimated standard deviation of the random effects for
percProm0is 3.650 (95% CrI: 3.167 to 4.229), indicating considerable variability in participants’ampl_sdat this prominence level.percProm1: The estimated standard deviation for
percProm1is 3.122 (95% CrI: 2.738 to 3.555), suggesting some variability in participants’ampl_sdat this level, although slightly lower than forpercProm0.percProm2: The estimated standard deviation for
percProm2is 2.985 (95% CrI: 2.612 to 3.417), indicating moderate variability inampl_sdat this prominence level, showing a further decrease compared topercProm1.percProm3: The estimated standard deviation for
percProm3is 2.877 (95% CrI: 2.479 to 3.372), suggesting that the variability inampl_sdis slightly less than atpercProm2, but still present across participants.
Overall Interpretation:
Trend: There is a gradual decrease in variability in
ampl_sdas thepercPromlevels increase from 0 to 3. This suggests that participants’ responses in terms ofampl_sdbecome slightly more consistent as the perceived prominence level increases.Implications: The relatively high standard deviations across all levels indicate that participants’ responses show substantial variability in
ampl_sd. However, the decrease in variability frompercProm0topercProm3suggests that participants may become slightly more uniform in theirampl_sdresponses as prominence increases.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.4 Amplitude median without silences (post-tonic)
4.2.4.1 Priors
4.2.4.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4915381 0.01902245 0.4536859 0.527662An R² of around 0.49 suggests a moderate fit. This
value means that the model explains approximately 49.2% of the variance
in the dependent variable. In other words, just under half of the
variability in ampl_noSilence_medianPost is accounted for
by the predictors in the model. The 95% credible interval indicates that
there is some uncertainty about the exact R² value, but it is likely
between 45.4% and 52.8%. This range suggests that the
model’s ability to explain the variance is relatively stable.
4.2.4.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: ampl_noSilence_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2277)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 3.36 0.27 2.88 3.93 1.00 2351
## sd(percProm1) 2.91 0.21 2.53 3.35 1.00 2192
## sd(percProm2) 2.65 0.20 2.29 3.09 1.00 2368
## sd(percProm3) 2.57 0.23 2.18 3.07 1.00 3285
## cor(percProm0,percProm1) 0.99 0.01 0.96 1.00 1.00 4136
## cor(percProm0,percProm2) 0.98 0.01 0.95 0.99 1.00 3063
## cor(percProm1,percProm2) 0.99 0.00 0.99 1.00 1.00 5511
## cor(percProm0,percProm3) 0.98 0.01 0.96 1.00 1.00 2998
## cor(percProm1,percProm3) 0.99 0.00 0.99 1.00 1.00 6687
## cor(percProm2,percProm3) 1.00 0.00 0.99 1.00 1.00 9274
## Tail_ESS
## sd(percProm0) 4593
## sd(percProm1) 4672
## sd(percProm2) 5014
## sd(percProm3) 6675
## cor(percProm0,percProm1) 6025
## cor(percProm0,percProm2) 5786
## cor(percProm1,percProm2) 9518
## cor(percProm0,percProm3) 6222
## cor(percProm1,percProm3) 10780
## cor(percProm2,percProm3) 12666
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 0.20 0.19 0.01 0.69 1.00 15637 9187
## percProm1 0.12 0.11 0.00 0.41 1.00 17903 8927
## percProm2 0.18 0.16 0.01 0.59 1.00 13404 9825
## percProm3 0.30 0.23 0.01 0.84 1.00 13389 9294
## gender_s 0.24 0.21 0.01 0.80 1.00 8207 7344
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.51 0.01 0.50 0.53 1.00 23873 11410
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 1.218554 1.204574 1.005292 1.991287
## percProm1 1.122730 1.118228 1.003054 1.503034
## percProm2 1.199982 1.170568 1.006013 1.803180
## percProm3 1.351038 1.252328 1.012274 2.326486
## gender_s 1.271736 1.239298 1.006549 2.221459It shows that ampl_noSilence_medianPost falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 0.00203 0.01508 0.02490 0.03119 0.04018 0.18309 674.2.4.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 1.17 1 1.79
## 1 1.10 1 1.42
## 2 1.17 1 1.67
## 3 1.31 1 2.11
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 0.05741 -0.437 0.782
## percProm0 - percProm2 0.00432 -0.675 0.793
## percProm0 - percProm3 -0.10798 -1.131 0.748
## percProm1 - percProm2 -0.05298 -0.653 0.370
## percProm1 - percProm3 -0.18026 -1.106 0.371
## percProm2 - percProm3 -0.11998 -0.989 0.523
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.4.3.2 Hypothesis testing
diff_0_1: Estimate = 0.082 [-0.287, 0.589], Posterior Probability = 64.2%; There is weak evidence that theampl_noSilence_medianPostatpercProm1is higher than atpercProm0. The credible interval includes zero, and the posterior probability is below 95%, so this difference is not considered reliable.diff_0_2: Estimate = 0.015 [-0.459, 0.559], Posterior Probability = 50.8%; There is no strong evidence for a difference betweenpercProm0andpercProm2, with the posterior probability close to chance.diff_0_3: Estimate = -0.103 [-0.706, 0.481], Posterior Probability = 64.8%; There is weak evidence that theampl_noSilence_medianPostatpercProm3is lower than atpercProm0. However, the credible interval includes zero, making the difference unreliable.diff_1_2: Estimate = -0.067 [-0.481, 0.282], Posterior Probability = 64.6%; There is weak evidence thatampl_noSilence_medianPostdecreases slightly frompercProm1topercProm2, but the result is not conclusive due to the wide credible interval.diff_1_3: Estimate = -0.185 [-0.747, 0.233], Posterior Probability = 77.8%; There is moderate evidence thatampl_noSilence_medianPostdecreases frompercProm1topercProm3. However, since the credible interval includes zero, the evidence remains inconclusive.diff_2_3: Estimate = -0.119 [-0.649, 0.336], Posterior Probability = 68.1%; There is weak evidence that theampl_noSilence_medianPostdecreases slightly frompercProm2topercProm3, but the result is not reliable due to the wide credible interval.
Overall Implications:
- Trend: There is no strong or consistent evidence of
significant differences between
percPromlevels for the variableampl_noSilence_medianPost. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and should be interpreted with caution.
4.2.4.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.4.4 Group-level effects
First, get a general overview.
percProm0: The estimated standard deviation of the random intercepts for
percProm0is 3.36 (95% CrI: 2.88 to 3.93), indicating a high level of variability in participants’ baselineampl_noSilence_medianPostat this level.percProm1: The estimated standard deviation for
percProm1is 2.91 (95% CrI: 2.53 to 3.35), showing moderate variability in participants’ response topercProm1. The range of uncertainty suggests that this variability is stable across participants.percProm2: The estimated standard deviation for
percProm2is 2.65 (95% CrI: 2.29 to 3.09), indicating that participants’ responses topercProm2are less variable compared topercProm0andpercProm1.percProm3: The estimated standard deviation for
percProm3is 2.57 (95% CrI: 2.18 to 3.07), suggesting a similar degree of variability aspercProm2, with slightly lower variability compared to the otherpercPromlevels.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.5 Amplitude median without silences
4.2.5.1 Priors
4.2.5.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5332912 0.01589941 0.5010413 0.5638605An R² of approximately 0.53 indicates a moderate
fit, meaning that the model explains about 53.3% of the
variance in the dependent variable,
ampl_noSilence_medianGer. In other words, just over half of
the variability in ampl_noSilence_medianGer is accounted
for by the model’s predictors. The 95% credible interval, ranging from
50.1% to 56.4%, suggests that the model’s explanatory
power is relatively stable, with some uncertainty around the exact
value. This indicates a solid but not overwhelming predictive
performance.
4.2.5.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: ampl_noSilence_median ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2344)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 3.23 0.26 2.77 3.78 1.00 1610
## sd(percProm1) 2.73 0.20 2.38 3.15 1.00 1413
## sd(percProm2) 2.63 0.20 2.27 3.06 1.00 1542
## sd(percProm3) 2.51 0.22 2.13 2.98 1.00 1979
## cor(percProm0,percProm1) 0.98 0.01 0.96 1.00 1.00 2496
## cor(percProm0,percProm2) 0.99 0.01 0.96 1.00 1.00 2318
## cor(percProm1,percProm2) 1.00 0.00 1.00 1.00 1.00 5830
## cor(percProm0,percProm3) 0.98 0.01 0.96 1.00 1.00 2498
## cor(percProm1,percProm3) 1.00 0.00 0.99 1.00 1.00 6599
## cor(percProm2,percProm3) 1.00 0.00 0.99 1.00 1.00 6514
## Tail_ESS
## sd(percProm0) 2511
## sd(percProm1) 2820
## sd(percProm2) 3372
## sd(percProm3) 3560
## cor(percProm0,percProm1) 4187
## cor(percProm0,percProm2) 4089
## cor(percProm1,percProm2) 9640
## cor(percProm0,percProm3) 5156
## cor(percProm1,percProm3) 10787
## cor(percProm2,percProm3) 10749
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 0.20 0.18 0.01 0.68 1.00 12925 7978
## percProm1 0.10 0.09 0.00 0.33 1.00 15398 8520
## percProm2 0.19 0.15 0.01 0.54 1.00 9577 8445
## percProm3 0.32 0.21 0.01 0.81 1.00 9075 6774
## gender_s 0.23 0.21 0.01 0.77 1.00 6052 6777
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.45 0.01 0.44 0.46 1.00 22738 11936
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 1.217194 1.202395 1.005051 1.967473
## percProm1 1.101149 1.094965 1.002825 1.395543
## percProm2 1.205558 1.156491 1.008038 1.716319
## percProm3 1.370609 1.238003 1.014996 2.242720
## gender_s 1.262965 1.227571 1.007569 2.153045It shows that ampl_noSilence_median falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.001484 0.017749 0.027888 0.033980 0.042728 0.1620214.2.5.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 1.17 1 1.78
## 1 1.09 1 1.33
## 2 1.18 1 1.62
## 3 1.34 1 2.06
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 0.06572 -0.333 0.777
## percProm0 - percProm2 -0.00519 -0.637 0.701
## percProm0 - percProm3 -0.14061 -1.112 0.674
## percProm1 - percProm2 -0.07997 -0.577 0.254
## percProm1 - percProm3 -0.23328 -1.022 0.250
## percProm2 - percProm3 -0.13573 -0.926 0.446
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.5.3.2 Hypothesis testing
diff_0_1: Estimate = 0.10 [-0.222, 0.598], Posterior Probability = 67.4%; There is weak evidence thatampl_noSilence_medianGeratpercProm1is higher than atpercProm0. The credible interval includes zero, and the posterior probability is low, suggesting this difference is not reliable.diff_0_2: Estimate = 0.01 [-0.421, 0.534], Posterior Probability = 48.9%; There is no strong evidence for a difference betweenpercProm0andpercProm2, with the estimate very close to zero and the posterior probability reflecting uncertainty.diff_0_3: Estimate = -0.119 [-0.682, 0.462], Posterior Probability = 68.3%; There is weak evidence thatampl_noSilence_medianGeratpercProm3is higher than atpercProm0, but the difference is not statistically significant, as the credible interval includes zero.diff_1_2: Estimate = -0.091 [-0.445, 0.192], Posterior Probability = 71.8%; There is weak evidence of a decrease inampl_noSilence_medianGerfrompercProm1topercProm2, but the interval includes zero, and the posterior probability does not reach a conclusive level.diff_1_3: Estimate = -0.219 [-0.714, 0.15], Posterior Probability = 84.8%; There is moderate evidence thatampl_noSilence_medianGeratpercProm1is lower than atpercProm3, but the evidence is not strong enough to be reliable.diff_2_3: Estimate = -0.128 [-0.615, 0.292], Posterior Probability = 70.6%; Weak evidence suggests thatampl_noSilence_medianGerincrease frompercProm2topercProm3, though this difference is not significant as the interval includes zero.
Overall Interpretation:
Trend: There is weak to moderate evidence of differences between prominence levels in
ampl_noSilence_medianGer, but none of the comparisons reach a high level of statistical significance (posterior probability > 95%).Interpretation: While some trends suggest a decrease in
ampl_noSilence_medianGeraspercPromincreases, the differences are not strong enough to draw reliable conclusions from these data.
4.2.5.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.5.4 Group-level effects
First, get a general overview.
percProm0: The estimated standard deviation of the random effects for
percProm0is 3.23 (95% CrI: 2.77 to 3.78), indicating substantial variability between participants at the lowest prominence level.percProm1: The estimated standard deviation for
percProm1is 2.73 (95% CrI: 2.38 to 3.15), suggesting notable variability in participants’ responses topercProm1. The variability is slightly lower than atpercProm0.percProm2: The estimated standard deviation for
percProm2is 2.63 (95% CrI: 2.27 to 3.06), indicating some decrease in variability compared topercProm1.percProm3: The estimated standard deviation for
percProm3is 2.51 (95% CrI: 2.13 to 2.98), suggesting that variability in participants’ responses topercProm3is the lowest among all prominence levels, but still substantial.
Implication: The variability in responses across
participants decreases as the perceived prominence level
(percProm) increases from percProm0 to
percProm3. However, there is still considerable individual
variability at all levels, particularly at the lower levels of
prominence.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.6 Duration
4.2.6.1 Priors
4.2.6.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.2538415 0.01672139 0.221985 0.2872682An R² of approximately 0.25 suggests a modest fit.
This value indicates that the model explains about 25.4% of the variance
in the dependent variable duration. In other words, a
quarter of the variability in duration is accounted for by
the predictors in the model. The 95% credible interval, ranging from
22.2% to 28.7%, shows that there is some uncertainty
around this estimate, but it provides a stable indication of the model’s
explanatory power.
4.2.6.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: duration ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2344)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 0.12 0.08 0.01 0.29 1.00 3362
## sd(percProm1) 0.12 0.02 0.08 0.16 1.00 8384
## sd(percProm2) 0.16 0.02 0.12 0.21 1.00 6206
## sd(percProm3) 0.24 0.04 0.18 0.31 1.00 7777
## cor(percProm0,percProm1) 0.16 0.35 -0.56 0.79 1.00 1222
## cor(percProm0,percProm2) 0.20 0.35 -0.54 0.79 1.00 1076
## cor(percProm1,percProm2) 0.86 0.08 0.66 0.98 1.00 6454
## cor(percProm0,percProm3) 0.14 0.37 -0.61 0.78 1.00 1263
## cor(percProm1,percProm3) 0.74 0.13 0.44 0.94 1.00 6158
## cor(percProm2,percProm3) 0.91 0.06 0.76 0.99 1.00 10445
## Tail_ESS
## sd(percProm0) 5055
## sd(percProm1) 11514
## sd(percProm2) 8900
## sd(percProm3) 10667
## cor(percProm0,percProm1) 2228
## cor(percProm0,percProm2) 1898
## cor(percProm1,percProm2) 9403
## cor(percProm0,percProm3) 2506
## cor(percProm1,percProm3) 9321
## cor(percProm2,percProm3) 13585
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 5.34 0.05 5.24 5.44 1.00 14198 11429
## percProm1 5.45 0.02 5.40 5.49 1.00 6290 9564
## percProm2 5.56 0.03 5.50 5.62 1.00 5490 8260
## percProm3 5.65 0.05 5.56 5.74 1.00 6540 9298
## gender_s 0.07 0.04 0.01 0.15 1.00 7867 6854
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.30 0.00 0.29 0.31 1.00 27809 11172
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 208.212541 1.052832 188.338451 230.704255
## percProm1 231.986769 1.023748 221.536232 243.057688
## percProm2 259.791602 1.029371 245.052276 274.977087
## percProm3 284.051734 1.046606 259.105665 310.231991
## gender_s 1.074742 1.038197 1.007181 1.163993It shows a stable increase from 0 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 69.0 205.0 256.0 271.0 315.2 895.04.2.6.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 208 188 230
## 1 232 221 243
## 2 260 245 275
## 3 285 260 311
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 -23.8 -47.4 -0.99
## percProm0 - percProm2 -51.7 -76.4 -26.98
## percProm0 - percProm3 -75.9 -109.2 -44.62
## percProm1 - percProm2 -27.8 -38.2 -17.48
## percProm1 - percProm3 -52.1 -74.2 -31.66
## percProm2 - percProm3 -24.4 -41.5 -8.27
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.6.3.2 Hypothesis testing
diff_0_1: Estimate = -0.108 [-0.217, 0.003], Posterior Probability = 97.3%; There is strong evidence that thedurationatpercProm1is higher than atpercProm0, with the credible interval slightly crossing zero.diff_0_2: Estimate = -0.221 [-0.332, -0.108], Posterior Probability = 100%; Strong evidence thatdurationatpercProm2is higher than atpercProm0, as indicated by the negative estimate and a credible interval entirely below zero.diff_0_3: Estimate = -0.311 [-0.441, -0.177], Posterior Probability = 100%; Strong evidence thatdurationatpercProm3is higher than atpercProm0.diff_1_2: Estimate = -0.113 [-0.154, -0.073], Posterior Probability = 100%; Strong evidence of an increase indurationfrompercProm1topercProm2.diff_1_3: Estimate = -0.202 [-0.276, -0.128], Posterior Probability = 100%; Strong evidence of an increase indurationfrompercProm1topercProm3.diff_2_3: Estimate = -0.089 [-0.146, -0.032], Posterior Probability = 99.8%; Strong evidence of an increase indurationfrompercProm2topercProm3.
Overall Implications:
Trend: The
durationincreases aspercPromlevels increase from 0 to 3.Interpretation: All comparisons, except for the borderline case of
diff_0_1, show strong evidence (posterior probability > 95%) of an increase indurationwith increasingpercProm. This suggests that higher prominence levels are associated with a longerduration.
4.2.6.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.6.4 Group-level effects
First, get a general overview.
percProm0: The estimated standard deviation of the random intercepts forpercProm0is 0.119 (95% CrI: 0.006 to 0.288), suggesting modest variability in participants’ baselinedurationat the lowest prominence level.percProm1: The estimated standard deviation is 0.116 (95% CrI: 0.083 to 0.157), indicating some variability in participants’ responses topercProm1.percProm2: The estimated standard deviation is 0.161 (95% CrI: 0.125 to 0.208), showing a slightly higher variability in response topercProm2compared topercProm1.percProm3: The estimated standard deviation is 0.237 (95% CrI: 0.177 to 0.315), indicating the most variability among participants at the highest prominence level (percProm3).
Implications:
- The variability in participants’
durationresponses increases with higher prominence levels, with the most substantial variation seen atpercProm3. This pattern suggests that participants’ realization of higher prominence levels is less uniform.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.7 Flux SD
4.2.7.1 Priors
4.2.7.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4174519 0.03402887 0.3503354 0.4827032An R² of approximately 0.42 suggests a moderate fit.
This value means that the model explains about 41.7% of the variance in
the flux_sd variable. In other words, a little over 40% of
the variability in flux_sd is accounted for by the
predictors in the model. The 95% credible interval indicates some
uncertainty about the exact R² value, but it is likely between
35.0% and 48.3%. This range shows that while the model
explains a significant portion of the variance, there is still room for
improvement in capturing all the variability in
flux_sd.
4.2.7.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: flux_sd ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2344)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 1.98 0.28 1.51 2.60 1.00 5075
## sd(percProm1) 2.22 0.22 1.84 2.72 1.00 5392
## sd(percProm2) 2.29 0.20 1.93 2.73 1.00 5069
## sd(percProm3) 2.47 0.22 2.07 2.92 1.00 5492
## cor(percProm0,percProm1) 0.80 0.09 0.59 0.92 1.00 2659
## cor(percProm0,percProm2) 0.78 0.09 0.56 0.92 1.00 2414
## cor(percProm1,percProm2) 0.99 0.00 0.98 1.00 1.00 7160
## cor(percProm0,percProm3) 0.78 0.09 0.55 0.92 1.00 2463
## cor(percProm1,percProm3) 0.99 0.01 0.97 1.00 1.00 7346
## cor(percProm2,percProm3) 1.00 0.00 0.99 1.00 1.00 9928
## Tail_ESS
## sd(percProm0) 8640
## sd(percProm1) 8770
## sd(percProm2) 8736
## sd(percProm3) 9116
## cor(percProm0,percProm1) 5444
## cor(percProm0,percProm2) 5150
## cor(percProm1,percProm2) 10693
## cor(percProm0,percProm3) 4488
## cor(percProm1,percProm3) 11060
## cor(percProm2,percProm3) 13294
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 0.25 0.23 0.01 0.87 1.00 7348 9628
## percProm1 0.23 0.18 0.01 0.68 1.00 10491 9880
## percProm2 0.14 0.13 0.00 0.48 1.00 15737 8107
## percProm3 0.11 0.11 0.00 0.40 1.00 18965 9098
## gender_s 0.19 0.18 0.01 0.67 1.00 14601 9404
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.92 0.01 0.90 0.95 1.00 27544 11856
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 1.286330 1.263735 1.007152 2.382555
## percProm1 1.253080 1.202646 1.008544 1.980475
## percProm2 1.151640 1.136491 1.003820 1.608740
## percProm3 1.115284 1.113929 1.003001 1.488405
## gender_s 1.214464 1.199854 1.005241 1.957386It shows that flux_sd systematically falls from 0 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0000145 0.0349014 0.0728800 0.0972661 0.1298478 0.48037074.2.7.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 1.21 1 2.09
## 1 1.21 1 1.83
## 2 1.12 1 1.51
## 3 1.09 1 1.39
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 0.0094 -0.829 1.078
## percProm0 - percProm2 0.0781 -0.579 1.055
## percProm0 - percProm3 0.1044 -0.397 1.166
## percProm1 - percProm2 0.0687 -0.439 0.764
## percProm1 - percProm3 0.0991 -0.358 0.834
## percProm2 - percProm3 0.0229 -0.389 0.488
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.7.3.2 Hypothesis testing
diff_0_1: Estimate = 0.026 [-0.528, 0.697], Posterior Probability = 51.70%; There is no strong evidence for a difference betweenflux_sdatpercProm0andpercProm1. The estimate is positive, but the credible interval includes zero, suggesting uncertainty in this comparison.diff_0_2: Estimate = 0.111 [-0.342, 0.755], Posterior Probability = 64.50%; There is moderate evidence thatflux_sdatpercProm2is higher than atpercProm0, but the evidence is not strong, as the credible interval still includes zero.diff_0_3: Estimate = 0.143 [-0.262, 0.786], Posterior Probability = 70.60%; There is moderate evidence thatflux_sdincreases frompercProm0topercProm3, but again, the credible interval includes zero, making the difference not conclusive.diff_1_2: Estimate = 0.084 [-0.303, 0.554], Posterior Probability = 65.50%; Moderate evidence suggests thatflux_sdincreases slightly frompercProm1topercProm2, but the credible interval includes zero, indicating uncertainty in the direction of the effect.diff_1_3: Estimate = 0.116 [-0.256, 0.597], Posterior Probability = 70.90%; There is moderate evidence thatflux_sdincreases frompercProm1topercProm3, but the evidence is not strong enough to confirm the increase.diff_2_3: Estimate = 0.032 [-0.293, 0.388], Posterior Probability = 57.70%; There is weak evidence for a difference betweenpercProm2andpercProm3. The credible interval includes zero, indicating considerable uncertainty.
Overall Implications:
Trend: While the estimates suggest a possible increase in
flux_sdaspercPromlevels increase, the credible intervals for most comparisons include zero, indicating that the evidence is not strong enough to confirm these differences.Interpretation: The comparisons do not show strong or consistent evidence of an increase in
flux_sdacross prominence levels. The posterior probabilities are all below 95%, indicating that none of the comparisons can be considered reliable.
4.2.7.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.7.4 Group-level effects
First, get a general overview.
percProm0: The estimated standard deviation of the random intercepts at
percProm0is 1.984 (95% CrI: 1.508 to 2.595), indicating notable variability in participants’flux_sdat this level.percProm1: The estimated standard deviation at
percProm1is 2.223 (95% CrI: 1.840 to 2.716), suggesting an increase in variability in participants’flux_sdcompared topercProm0.percProm2: The estimated standard deviation at
percProm2is 2.294 (95% CrI: 1.925 to 2.731), indicating a slight increase in variability compared topercProm1.percProm3: The estimated standard deviation at
percProm3is 2.466 (95% CrI: 2.073 to 2.924), showing further increased variability in participants’flux_sdcompared to the other prominence levels.
Interpretation: The increasing standard deviations
across percProm levels suggest that variability in
participants’ flux_sd tends to grow as perceived prominence
increases. This implies greater individual differences in how
participants’ flux_sd changes across different levels of
prominence.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.8 Flux median
4.2.8.1 Priors
4.2.8.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4894135 0.008934676 0.470584 0.5019241An R² of 0.489 (95% CrI: 0.470 to 0.501) indicates a moderate
fit, meaning that approximately 48.9% of the
variance in flux_median is explained by the model. This
credible interval suggests a stable estimate within a range of explained
variability.
4.2.8.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: flux_median ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2344)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 5.42 0.41 4.67 6.26 1.00 6848
## sd(percProm1) 3.68 0.26 3.20 4.25 1.00 6845
## sd(percProm2) 3.25 0.22 2.85 3.72 1.00 6385
## sd(percProm3) 3.13 0.24 2.70 3.65 1.00 7524
## cor(percProm0,percProm1) 0.79 0.07 0.63 0.90 1.00 4236
## cor(percProm0,percProm2) 0.79 0.07 0.64 0.90 1.00 3921
## cor(percProm1,percProm2) 0.97 0.01 0.94 0.99 1.00 7237
## cor(percProm0,percProm3) 0.81 0.07 0.64 0.92 1.00 4447
## cor(percProm1,percProm3) 0.94 0.03 0.88 0.98 1.00 8466
## cor(percProm2,percProm3) 0.98 0.01 0.95 1.00 1.00 12206
## Tail_ESS
## sd(percProm0) 9794
## sd(percProm1) 8985
## sd(percProm2) 8707
## sd(percProm3) 10362
## cor(percProm0,percProm1) 7222
## cor(percProm0,percProm2) 6422
## cor(percProm1,percProm2) 10596
## cor(percProm0,percProm3) 8103
## cor(percProm1,percProm3) 10931
## cor(percProm2,percProm3) 14385
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 0.28 0.26 0.01 0.95 1.00 14295 9081
## percProm1 0.25 0.23 0.01 0.87 1.00 16139 9420
## percProm2 0.16 0.16 0.00 0.57 1.00 19134 8938
## percProm3 0.20 0.19 0.01 0.69 1.00 16999 9742
## gender_s 0.28 0.26 0.01 0.98 1.00 18520 9927
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 2.37 0.04 2.30 2.44 1.00 26057 11789
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 1.323592 1.295039 1.007500 2.574693
## percProm1 1.280032 1.262147 1.006795 2.384896
## percProm2 1.175126 1.169967 1.004069 1.772865
## percProm3 1.221109 1.207878 1.005733 1.995472
## gender_s 1.325376 1.300302 1.007319 2.677204It shows that flux_median falls from 0 to 1 and 1 to 2, but raises from 2 to 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 8.000e-08 4.676e-03 8.396e-03 1.397e-02 1.600e-02 2.274e-014.2.8.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 1.25 1 2.26
## 1 1.22 1 2.07
## 2 1.14 1 1.64
## 3 1.18 1 1.82
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 0.0242 -1.141 1.346
## percProm0 - percProm2 0.0868 -0.652 1.311
## percProm0 - percProm3 0.0544 -0.815 1.362
## percProm1 - percProm2 0.0584 -0.657 1.111
## percProm1 - percProm3 0.0320 -0.856 1.138
## percProm2 - percProm3 -0.0236 -0.844 0.627
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.8.3.2 Hypothesis testing
diff_0_1: Estimate =
0.033[-0.670, 0.777], Posterior Probability = 53.7%; There is no strong evidence for a difference betweenpercProm0andpercProm1, with the posterior probability close to chance.diff_0_2: Estimate =
0.119[-0.415, 0.829], Posterior Probability = 64.0%; There is weak evidence suggesting thatflux_medianatpercProm2is higher than atpercProm0, though the credible interval includes zero, making the difference unreliable.diff_0_3: Estimate =
0.081[-0.521, 0.807], Posterior Probability = 58.6%; No strong evidence for a difference betweenpercProm0andpercProm3, with a credible interval spanning zero.diff_1_2: Estimate =
0.086[-0.418, 0.741], Posterior Probability = 60.8%; Weak evidence indicates thatflux_medianmay increase frompercProm1topercProm2, though the result is inconclusive.diff_1_3: Estimate =
0.047[-0.539, 0.723], Posterior Probability = 55.5%; There is no strong evidence for a difference betweenpercProm1andpercProm3, given the credible interval includes zero.diff_2_3: Estimate =
-0.038[-0.569, 0.441], Posterior Probability = 55.7%; There is weak evidence of a slight decrease influx_medianfrompercProm2topercProm3, but the difference is unreliable.
Overall Implications:
- Trend: There is no strong or consistent evidence of
significant differences between
percPromlevels forflux_median. The posterior probabilities and wide credible intervals suggest that any potential differences are uncertain and should be interpreted cautiously.
4.2.8.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.8.4 Group-level effects
First, get a general overview.
percProm0: SD = 1.984 (95% CrI: 1.508 to 2.595), indicating moderate variability in participants’
flux_median.percProm1: SD = 2.223 (95% CrI: 1.840 to 2.716), suggesting an increase in variability compared to
percProm0.percProm2: SD = 2.294 (95% CrI: 1.925 to 2.731), indicating further increased variability.
percProm3: SD = 2.466 (95% CrI: 2.073 to 2.924), showing the highest variability, indicating substantial differences across participants in
flux_median.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.9 Amplitude SD without silences
4.2.9.1 Priors
4.2.9.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.5297911 0.01536498 0.4984655 0.5587921An R² of 0.530 (95% CrI: 0.498 to 0.559) indicates a moderate
fit, with approximately 53% of the variance in
ampl_noSilence_sd explained by the model. The narrow
credible interval suggests stability in the estimate.
4.2.9.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: ampl_noSilence_sd ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 2344)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 3.67 0.27 3.18 4.26 1.00 2276
## sd(percProm1) 3.14 0.21 2.75 3.59 1.00 1975
## sd(percProm2) 3.01 0.21 2.62 3.45 1.00 2145
## sd(percProm3) 2.91 0.23 2.50 3.41 1.00 2773
## cor(percProm0,percProm1) 0.98 0.01 0.96 0.99 1.00 3876
## cor(percProm0,percProm2) 0.98 0.01 0.95 0.99 1.00 3335
## cor(percProm1,percProm2) 1.00 0.00 1.00 1.00 1.00 6795
## cor(percProm0,percProm3) 0.97 0.01 0.94 0.99 1.00 3177
## cor(percProm1,percProm3) 0.99 0.00 0.99 1.00 1.00 6122
## cor(percProm2,percProm3) 1.00 0.00 0.99 1.00 1.00 7055
## Tail_ESS
## sd(percProm0) 4497
## sd(percProm1) 4288
## sd(percProm2) 4298
## sd(percProm3) 5632
## cor(percProm0,percProm1) 6122
## cor(percProm0,percProm2) 5482
## cor(percProm1,percProm2) 11049
## cor(percProm0,percProm3) 5580
## cor(percProm1,percProm3) 9943
## cor(percProm2,percProm3) 12196
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 0.23 0.21 0.01 0.78 1.00 16189 9946
## percProm1 0.12 0.11 0.00 0.39 1.00 19139 9704
## percProm2 0.18 0.14 0.01 0.53 1.00 15145 8849
## percProm3 0.27 0.22 0.01 0.80 1.00 17008 9829
## gender_s 0.24 0.22 0.01 0.81 1.00 10463 7318
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.48 0.01 0.46 0.49 1.00 29045 11764
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 1.263459 1.237401 1.006980 2.180809
## percProm1 1.122085 1.112960 1.003245 1.482347
## percProm2 1.200015 1.153557 1.007086 1.701875
## percProm3 1.309910 1.240539 1.010086 2.231745
## gender_s 1.276043 1.246767 1.006776 2.238174It shows that ampl_noSilence_sd falls from 0 to 1 and 1 to 2, but raises from 2 to 3 (surpassing 0).
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.0006497 0.0073900 0.0120082 0.0150229 0.0192005 0.07959824.2.9.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 1.21 1 1.97
## 1 1.10 1 1.42
## 2 1.18 1 1.61
## 3 1.26 1 2.05
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 0.0855 -0.426 0.947
## percProm0 - percProm2 0.0257 -0.618 0.907
## percProm0 - percProm3 -0.0385 -1.064 0.926
## percProm1 - percProm2 -0.0605 -0.566 0.344
## percProm1 - percProm3 -0.1406 -1.052 0.358
## percProm2 - percProm3 -0.0754 -0.850 0.542
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.9.3.2 Hypothesis testing
diff_0_1: Estimate =
0.119[-0.271, 0.685], Posterior Probability = 68.2%; There is weak evidence thatampl_noSilence_sdatpercProm1is higher than atpercProm0, but the result is not conclusive.diff_0_2: Estimate =
0.052[-0.401, 0.649], Posterior Probability = 55.0%; There is no strong evidence for a difference betweenpercProm0andpercProm2, with the posterior probability close to chance.diff_0_3: Estimate =
-0.036[-0.650, 0.595], Posterior Probability = 55.6%; There is no conclusive evidence of a difference betweenpercProm0andpercProm3, given the wide credible interval.diff_1_2: Estimate =
-0.067[-0.427, 0.261], Posterior Probability = 66.6%; Weak evidence suggests a slight decrease inampl_noSilence_sdfrompercProm1topercProm2, though this difference remains unreliable.diff_1_3: Estimate =
-0.155[-0.707, 0.251], Posterior Probability = 74.5%; Moderate evidence of a decrease inampl_noSilence_sdfrompercProm1topercProm3, but it is not conclusive due to the inclusion of zero.diff_2_3: Estimate =
-0.088[-0.606, 0.329], Posterior Probability = 63.8%; Weak evidence thatampl_noSilence_sddecreases frompercProm2topercProm3, though the result is not reliable.
Overall Implications:
- Trend: There is no consistent or strong evidence of
significant differences between
percPromlevels forampl_noSilence_sd. The posterior probabilities and wide credible intervals imply that any potential differences are uncertain and should be interpreted with caution.
4.2.9.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.9.4 Group-level effects
First, get a general overview.
percProm0: SD = 3.672 (95% CrI: 3.183 to 4.263), indicating high variability in participants’
ampl_noSilence_sd.percProm1: SD = 3.142 (95% CrI: 2.752 to 3.587), suggesting reduced variability.
percProm2: SD = 3.009 (95% CrI: 2.625 to 3.451), indicating a slight decrease in variability compared to
percProm1.percProm3: SD = 2.905 (95% CrI: 2.504 to 3.408), showing the lowest variability, suggesting participants’ responses are more consistent at this level.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
4.2.10 F0 median (post-tonic)
4.2.10.1 Priors
4.2.10.2 Model
Calculate R².
## Estimate Est.Error Q2.5 Q97.5
## R2 0.4325174 0.01226329 0.4081115 0.4560081An R² of 0.433 (95% CrI: 0.408 to 0.456) suggests a moderate
fit, with approximately 43.3% of the variance
in pitch_medianPost explained by the model. The interval
implies a stable estimate within the range.
4.2.10.3 Fixed effects
Let’s check how it looks like.
## Family: lognormal
## Links: mu = identity; sigma = identity
## Formula: pitch_medianPost ~ 0 + percProm + gender_s + (0 + percProm | participant)
## Data: data_prepost_ger (Number of observations: 1821)
## Draws: 4 chains, each with iter = 8000; warmup = 4000; thin = 1;
## total post-warmup draws = 16000
##
## Multilevel Hyperparameters:
## ~participant (Number of levels: 35)
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS
## sd(percProm0) 0.13 0.06 0.02 0.27 1.00 5473
## sd(percProm1) 0.09 0.02 0.07 0.13 1.00 9059
## sd(percProm2) 0.08 0.01 0.06 0.11 1.00 7520
## sd(percProm3) 0.08 0.02 0.05 0.12 1.00 9609
## cor(percProm0,percProm1) 0.30 0.33 -0.42 0.85 1.00 1605
## cor(percProm0,percProm2) 0.14 0.37 -0.58 0.79 1.00 1322
## cor(percProm1,percProm2) 0.68 0.14 0.36 0.90 1.00 6866
## cor(percProm0,percProm3) -0.02 0.40 -0.74 0.73 1.00 1965
## cor(percProm1,percProm3) 0.55 0.20 0.12 0.87 1.00 9960
## cor(percProm2,percProm3) 0.70 0.16 0.33 0.94 1.00 10002
## Tail_ESS
## sd(percProm0) 4012
## sd(percProm1) 11703
## sd(percProm2) 11132
## sd(percProm3) 9856
## cor(percProm0,percProm1) 2687
## cor(percProm0,percProm2) 3253
## cor(percProm1,percProm2) 10418
## cor(percProm0,percProm3) 4592
## cor(percProm1,percProm3) 11718
## cor(percProm2,percProm3) 12171
##
## Regression Coefficients:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## percProm0 5.13 0.05 5.03 5.24 1.00 7530 10205
## percProm1 5.13 0.02 5.10 5.17 1.00 8390 9919
## percProm2 5.21 0.01 5.18 5.24 1.00 9141 10531
## percProm3 5.21 0.02 5.17 5.25 1.00 10635 12435
## gender_s 0.33 0.03 0.28 0.39 1.00 9148 10611
##
## Further Distributional Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma 0.18 0.00 0.17 0.18 1.00 21258 11305
##
## Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).## Estimate Est.Error Q2.5 Q97.5
## percProm0 169.820804 1.054882 152.182838 188.107643
## percProm1 169.526200 1.018666 163.403936 175.762420
## percProm2 183.921112 1.015061 178.553955 189.478645
## percProm3 183.263705 1.019559 176.436022 190.420013
## gender_s 1.393311 1.027689 1.321521 1.469917It shows that pitch_medianPost is the same on 0 and 1, and raises to stay the same on 2 and 3.
Diagnostic plots.
Posterior predictive check.
Extract samples.
Check conditional effect.
Compare conditional effects to raw values.
## Min. 1st Qu. Median Mean 3rd Qu. Max. NA's
## 90.61 163.13 192.13 196.89 227.88 450.43 5234.2.10.3.1 Effect comparison
## percProm emmean lower.HPD upper.HPD
## 0 172 154 190
## 1 172 166 178
## 2 186 181 192
## 3 186 179 193
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
## contrast estimate lower.HPD upper.HPD
## percProm0 - percProm1 0.370 -19.08 18.20
## percProm0 - percProm2 -14.105 -32.15 4.95
## percProm0 - percProm3 -13.421 -33.23 5.73
## percProm1 - percProm2 -14.610 -20.25 -9.28
## percProm1 - percProm3 -13.932 -21.39 -6.40
## percProm2 - percProm3 0.629 -5.63 7.19
##
## Results are averaged over the levels of: gender_s
## Point estimate displayed: median
## HPD interval probability: 0.95
If the comparison includes 0, the contrast is not reliably different.
4.2.10.3.2 Hypothesis testing
diff_0_1: Estimate =
0.002[-0.109, 0.109], Posterior Probability = 51.7%; No strong evidence for a difference betweenpercProm0andpercProm1, with a posterior probability close to chance.diff_0_2: Estimate =
-0.08[-0.190, 0.024], Posterior Probability = 93.5%; Moderate evidence thatpitch_medianPostatpercProm0is slightly lower than atpercProm2, but not conclusive as the credible interval includes zero.diff_0_3: Estimate =
-0.076[-0.191, 0.033], Posterior Probability = 91.9%; Moderate evidence suggests a slight decrease inpitch_medianPostatpercProm3compared topercProm0, though it remains inconclusive.diff_1_2: Estimate =
-0.081[-0.113, -0.05], Posterior Probability = 100%; Strong evidence of a decrease inpitch_medianPostfrompercProm1topercProm2, as the credible interval lies entirely below zero.diff_1_3: Estimate =
-0.078[-0.119, -0.036], Posterior Probability = 100%; Strong evidence thatpitch_medianPostatpercProm1is lower than atpercProm3, with a negative estimate and high posterior probability.diff_2_3: Estimate =
0.004[-0.031, 0.038], Posterior Probability = 58.1%; No strong evidence for a difference betweenpercProm2andpercProm3.
Overall Implications:
- Trend: There is strong evidence of a decreasing
trend in
pitch_medianPostfrompercProm1to bothpercProm2andpercProm3. While the comparisons involvingpercProm0are inconclusive, these findings suggest that as perceived prominence increases,pitch_medianPostgenerally decreases across most levels.
4.2.10.3.3 Plot fixef
Visualize the (non-)linearity of the effect with a violin plot of the posteriors for each prominence level and a GAMM overlaid on top.
4.2.10.4 Group-level effects
First, get a general overview.
percProm0: SD = 0.133 (95% CrI: 0.023 to 0.267), indicating minor variability in
pitch_medianPost.percProm1: SD = 0.092 (95% CrI: 0.065 to 0.125), suggesting lower variability in response at
percProm1.percProm2: SD = 0.079 (95% CrI: 0.059 to 0.105), indicating a further decrease in variability compared to
percProm1.percProm3: SD = 0.083 (95% CrI: 0.050 to 0.123), showing slightly more variability than
percProm2.
Extract samples in interpretable units. Then, plot them so that we can see how each participant uses the levels of percProm.
First, we extract individual draws. Here, if the CrI encompasses 0, the participant does not differ from the overall behavior on this percProm. Then, we plot random effects for each participant and percProm level on the model scale, representing deviations from the fixed effects.
This concludes the analysis.
5 Session info
sessionInfo()## R version 4.4.1 (2024-06-14 ucrt)
## Platform: x86_64-w64-mingw32/x64
## Running under: Windows 10 x64 (build 19045)
##
## Matrix products: default
##
##
## locale:
## [1] LC_COLLATE=German_Germany.utf8 LC_CTYPE=German_Germany.utf8
## [3] LC_MONETARY=German_Germany.utf8 LC_NUMERIC=C
## [5] LC_TIME=German_Germany.utf8
##
## time zone: Europe/Berlin
## tzcode source: internal
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] distributional_0.5.0 gganimate_1.0.9 RColorBrewer_1.1-3
## [4] ggrepel_0.9.6 rstan_2.32.6 StanHeaders_2.32.10
## [7] cowplot_1.1.3 tidybayes_3.0.7 ggdist_3.3.2
## [10] modelr_0.1.11 magrittr_2.0.3 posterior_1.6.0
## [13] emmeans_1.10.5 cmdstanr_0.8.1 brms_2.22.0
## [16] Rcpp_1.0.13 corrplot_0.94 gridExtra_2.3
## [19] ggpubr_0.6.0 ggforce_0.4.2 data.table_1.16.0
## [22] lubridate_1.9.3 forcats_1.0.0 dplyr_1.1.4
## [25] purrr_1.0.2 readr_2.1.5 tidyr_1.3.1
## [28] ggplot2_3.5.1 tidyverse_2.0.0 stringr_1.5.1
## [31] tibble_3.2.1
##
## loaded via a namespace (and not attached):
## [1] inline_0.3.19 rlang_1.1.4 matrixStats_1.4.1
## [4] compiler_4.4.1 mgcv_1.9-1 loo_2.8.0
## [7] systemfonts_1.1.0 reshape2_1.4.4 vctrs_0.6.5
## [10] crayon_1.5.3 pkgconfig_2.0.3 arrayhelpers_1.1-0
## [13] fastmap_1.2.0 backports_1.5.0 labeling_0.4.3
## [16] utf8_1.2.4 rmarkdown_2.28 tzdb_0.4.0
## [19] ps_1.8.0 ragg_1.3.3 bit_4.5.0
## [22] xfun_0.48 cachem_1.1.0 jsonlite_1.8.9
## [25] progress_1.2.3 highr_0.11 tweenr_2.0.3
## [28] prettyunits_1.2.0 broom_1.0.7 parallel_4.4.1
## [31] R6_2.5.1 bslib_0.8.0 stringi_1.8.4
## [34] car_3.1-3 jquerylib_0.1.4 estimability_1.5.1
## [37] knitr_1.48 bayesplot_1.11.1 splines_4.4.1
## [40] Matrix_1.7-0 timechange_0.3.0 tidyselect_1.2.1
## [43] rstudioapi_0.16.0 abind_1.4-8 yaml_2.3.10
## [46] codetools_0.2-20 processx_3.8.4 pkgbuild_1.4.4
## [49] plyr_1.8.9 lattice_0.22-6 withr_3.0.1
## [52] bridgesampling_1.1-2 coda_0.19-4.1 evaluate_1.0.0
## [55] RcppParallel_5.1.9 polyclip_1.10-7 pillar_1.9.0
## [58] carData_3.0-5 tensorA_0.36.2.1 checkmate_2.3.2
## [61] stats4_4.4.1 generics_0.1.3 vroom_1.6.5
## [64] hms_1.1.3 rstantools_2.4.0 munsell_0.5.1
## [67] scales_1.3.0 xtable_1.8-4 glue_1.8.0
## [70] tools_4.4.1 ggsignif_0.6.4 mvtnorm_1.3-1
## [73] grid_4.4.1 QuickJSR_1.4.0 colorspace_2.1-1
## [76] nlme_3.1-164 Formula_1.2-5 cli_3.6.3
## [79] textshaping_0.4.0 fansi_1.0.6 svUnit_1.0.6
## [82] Brobdingnag_1.2-9 gtable_0.3.5 rstatix_0.7.2
## [85] sass_0.4.9 digest_0.6.37 farver_2.1.2
## [88] htmltools_0.5.8.1 lifecycle_1.0.4 bit64_4.5.2
## [91] MASS_7.3-60.2